Consequences of Continuity
|
|
- Grant Evans
- 5 years ago
- Views:
Transcription
1 Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017
2 Outline 1 Domains of Continuous Functions 2 The Intermediate Value Theorem 3 Consequences of a compact domain for a continuous function 4 Continuity in terms of inverse images of open sets
3 Domains of Continuous Functions In our discussions of continuity, we always assume that f is defined locally at the point p which means there is a radius r so that if x B r (p), f (x) is defined.
4 Domains of Continuous Functions In our discussions of continuity, we always assume that f is defined locally at the point p which means there is a radius r so that if x B r (p), f (x) is defined. If at each point p in dom(f ), f is locally defined, this says each such p is an interior point of dom(f ) and so dom(f ) is an open set.
5 Domains of Continuous Functions In our discussions of continuity, we always assume that f is defined locally at the point p which means there is a radius r so that if x B r (p), f (x) is defined. If at each point p in dom(f ), f is locally defined, this says each such p is an interior point of dom(f ) and so dom(f ) is an open set. Because of the requirement of f being locally defined at each p, we see dom(f ) can not have isolated boundary points. Hence, any boundary points dom(f ) has must be accumulation points. This means if p (dom(f )), there has to be a sequence (x n ) in dom(f ), each x n p, with x n p.
6 Domains of Continuous Functions In our discussions of continuity, we always assume that f is defined locally at the point p which means there is a radius r so that if x B r (p), f (x) is defined. If at each point p in dom(f ), f is locally defined, this says each such p is an interior point of dom(f ) and so dom(f ) is an open set. Because of the requirement of f being locally defined at each p, we see dom(f ) can not have isolated boundary points. Hence, any boundary points dom(f ) has must be accumulation points. This means if p (dom(f )), there has to be a sequence (x n ) in dom(f ), each x n p, with x n p. If for any sequence like this, we know f (x n ) a for some value a, this implies lim x p f (x) exists and equals a. Hence, we can define f (p) = a and f therefore has a removeable discontinuity at such a p.
7 Domains of Continuous Functions Now subsets of R can be very complicated, so let s restrict our attention to intervals. An interval of R is a subset I of the following form I = (, a) infinite open interval I = {, a} I = (, a] infinite closed interval I = {, a} I = (a, ) infinite open interval I = {a,, } I = [a, ) infinite closed interval I = {a,, } I = (, ) infinite open and closed interval I = {, } I = (a, b), a < b finite open interval I = {a, b} I = [a, b), a < b finite half open interval I = {a, b} I = (a, b], a < b finite half open interval I = {a, b} I = [a, b], a < b finite closed interval I = {a, b}
8 Domains of Continuous Functions Now subsets of R can be very complicated, so let s restrict our attention to intervals. An interval of R is a subset I of the following form I = (, a) infinite open interval I = {, a} I = (, a] infinite closed interval I = {, a} I = (a, ) infinite open interval I = {a,, } I = [a, ) infinite closed interval I = {a,, } I = (, ) infinite open and closed interval I = {, } I = (a, b), a < b finite open interval I = {a, b} I = [a, b), a < b finite half open interval I = {a, b} I = (a, b], a < b finite half open interval I = {a, b} I = [a, b], a < b finite closed interval I = {a, b} Continuity at the points in I is possible but not guaranteed. Examples: 1/x is continuous on (0, ) and the boundary point 0 can not be included. x sin(1/x) on (0, 1] can be extended to the boundary point 0 so that x sin(1/x) has a removeable discontinuity at 0. It is possible to talk about boundary points at ± but we won t do that here.
9 The Intermediate Value Theorem Theorem The Intermediate Value Theorem for [a, b] Let f : [a, b] R be continuous with f (a) f (b). Then if y is between f (a) and f (b), there is at least one c between a and b so that f (c) = y. Proof For concreteness, let s assume f (a) < f (b). We then have f (a) < y < f (b) and we want to find a c so that a < c < b and f (c) = y. Our argument here will be very abstract! So lot s of fun! Let A = {x [a, b] : f (x) y}. Since f (a) < y we know A is non empty. Further, since x = b is the largest value allowed, we see A is bounded. So by the completeness axiom, sup(a) exists and is finite. Let c = sup(a). Since b is an upper bound of A, we see by the definition of supremum, that c b.
10 The Intermediate Value Theorem Proof We now show f (c) = y. (Step 1: f (c) y): (Case (a) ): if c A, then by the definition of A, we must have f (c) y. (Case (b) ): Assume c A. Then applying the Supremum Tolerance Lemma using the sequence of tolerances ɛ = 1/n, we can construct a sequence (x n ) with c 1/n < x n c and x n A. Since we are in Case (b), none of these x n can be c. Since f is continuous at c, we then must have lim n f (x n ) = f (c). But since x n A, we also must have f (x n ) y for all n. In general, if a n a and b n b with a n b n, this implies a b. To see this, note for any ɛ > 0, there is an N 1 so that n > N 1 implies a ɛ/2 < a n < a + ɛ/2 and there is an N 2 so that n > N 2 implies b ɛ/2 < b n < b + ɛ/2. So for n > max{n 1, N 2 }, we have a ɛ/2 < a n b n < b + ɛ/2. But this tells us that a b < ɛ.
11 The Intermediate Value Theorem Proof Now if a > b, we could choose ɛ = (a b)/2 and then we would have a b < (a b)/2 implying (a b)/2 < 0 which contradicts our assumption. Hence, we must have a b. Applying this result here, f (x n ) y for all n thus implies f (c) y. So in either Case (a) or Case (b). f (c) y. (Step 2: f (c) = y): Assume f (c) < y. Since we know y < f (b), this says c < b. Choose ɛ = (1/2)(y f (c)) > 0. Since f is continuous at c, for this ɛ, there is a δ > 0 so that B δ (c) [a, b] and x c < δ f (x) f (c) < ɛ. Rewriting, we have c δ < x < c + δ f (c) ɛ < f (x) < f (c) + ɛ Since B δ (c) [a, b], we know (c, c + δ) (c, b]. Now pick a point z in (c, c + δ). Then, f (z) < f (c) + ɛ.
12 The Intermediate Value Theorem Proof But ɛ = (1/2)(y f (c)). Thus, f (z) < (y + f (c))/2. Now we assumed f (c) < y so the average (f (c) + y)/2 < y also. So we have f (z) < y. But this says z A with z > c which is the supremum of A. This is not possible and so our assumption that f (c) < y is wrong and we must have f (c) = y. Theorem The Intermediate Value Theorem for Intervals Let f : I R be continuous. Let a and b be any two points in I with a b and f (a) f (b). Then if y is between f (a) and f (b), there is at least one c between a and b so that f (c) = y.
13 The Intermediate Value Theorem Proof Apply the Intermediate Value Theorem for [a, b] I and the result follows. Theorem Let I be an interval and assume f is continuous on I. Then f (I ) is also an interval. Proof Let u, v be in f (I ) with u < v. Then there are s, t I so that f (s) = u and f (t) = v. Choose any number y so that u < y < v. (Case (a)): If s < t, since f is continuous on [s, t], by the Intermediate Value Theorem for finite closes intervals, there is a number c with s < c < t with f (c) = y. Thus, y f (I ).
14 The Intermediate Value Theorem Proof (Case (b)) If t < s, we apply the same sort of argument to again show y f (I ). Combining these arguments, we see for any u, v f (I ) with u < v, all u < y < v are also in f (I ). This shows f (I ) is an interval.
15 Consequences of a compact domain for a continuous function What if the domain of a continuous function f is compact? What are the consequences? Theorem If f : dom(f ) R with dom(f ) a compact set, then f (dom(f )) is a compact set also. Proof For convenience, let K = dom(f ). Then f (K) = {f (x) : x K}. Let (y n ) be any sequence in f (K). Then there is a sequence (x n ) K with y n = f (x n ) for all n. Since K is compact, there is a subsequence (x 1 n ) of (x n ) so that x 1 n p for some p K. Since f is continuous at p, we then have f (x 1 n ) f (p). ( Recall at a boundary point of K, we would interpret continuity at this point in terms of right or left continuity). Hence (y n ) has a subsequence (y 1 n ) = (f (x 1 n )) which converges to an element of f (K). Thus, f (K) is sequentially compact and topologically compact and closed and bounded.
16 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum.
17 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum. It is a straightforward argument to show there are sequences (x m n ) and (x M n ) so that x m n inf(ω) and x M n sup(ω).
18 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum. It is a straightforward argument to show there are sequences (x m n ) and (x M n ) so that x m n inf(ω) and x M n sup(ω). If the inf(ω) Ω, then we say the minimum of the set Ω is achieved by at least the point x m = inf(ω) and xn m x m. This is also called the absolute minimum of Ω.
19 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum. It is a straightforward argument to show there are sequences (x m n ) and (x M n ) so that x m n inf(ω) and x M n sup(ω). If the inf(ω) Ω, then we say the minimum of the set Ω is achieved by at least the point x m = inf(ω) and xn m x m. This is also called the absolute minimum of Ω. If the sup(ω) Ω, then we say the maximum of the set Ω is achieved by at least the point x M = sup(ω) and xn M x M. This is also called the absolute maximum of Ω.
20 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum. It is a straightforward argument to show there are sequences (x m n ) and (x M n ) so that x m n inf(ω) and x M n sup(ω). If the inf(ω) Ω, then we say the minimum of the set Ω is achieved by at least the point x m = inf(ω) and xn m x m. This is also called the absolute minimum of Ω. If the sup(ω) Ω, then we say the maximum of the set Ω is achieved by at least the point x M = sup(ω) and xn M x M. This is also called the absolute maximum of Ω. So Ω compact implies Ω always has an absolute minimum and maximum value.
21 Consequences of a compact domain for a continuous function Let s recall what we know about minima and maxima of sets. First if a set Ω is nonempty and bounded, it has a finite infimum and supremum. It is a straightforward argument to show there are sequences (x m n ) and (x M n ) so that x m n inf(ω) and x M n sup(ω). If the inf(ω) Ω, then we say the minimum of the set Ω is achieved by at least the point x m = inf(ω) and xn m x m. This is also called the absolute minimum of Ω. If the sup(ω) Ω, then we say the maximum of the set Ω is achieved by at least the point x M = sup(ω) and xn M x M. This is also called the absolute maximum of Ω. So Ω compact implies Ω always has an absolute minimum and maximum value. If K is compact with f continuous, f (K) is compact. Thus, f (K) has an absolute minimum and maximum; i.e. there exist x m, x M K f (x m ) f (x) and f (x M ) f (x) x K. The points x m and x M need not be unique, of course.
22 Consequences of a compact domain for a continuous function Here are more consequences. If f : [a, b] R is continuous on [a, b], first note this means f is left continuous at b and right continuous at a. Note, to use full circles, we could redefine f like this: f (a), x < a ˆf (x) = f (x), x [a, b] f (b), x > b i.e. we extend f to the right and left based on the f (a) and f (b) value. Then f is continuous at a and b in the usual sense. We don t usually need to be so precise however.
23 Consequences of a compact domain for a continuous function Here are more consequences. If f : [a, b] R is continuous on [a, b], first note this means f is left continuous at b and right continuous at a. Note, to use full circles, we could redefine f like this: f (a), x < a ˆf (x) = f (x), x [a, b] f (b), x > b i.e. we extend f to the right and left based on the f (a) and f (b) value. Then f is continuous at a and b in the usual sense. We don t usually need to be so precise however. So from what we have said f ([a, b]) is compact when f is continuous on K. Thus, f ([a, b]) must have a minimum and maximum value, m and M so that f (x) M = f (x M ) and f (x) f (x m ) for all x [a, b].
24 Consequences of a compact domain for a continuous function Theorem Let f : [a, b] R be continuous. Then f ([a, b]) = [f (x m ), f (x M )] where x m, x M are in [a, b] and f (x m ) and f (x M ) are the absolute minimum and maximum of f on [a, b], respectively. Proof We already know f ([a, b]) is an interval and it is a compact set. and we have f (x m ) f (x) f (x M ) for all x [a, b]. Thus f ([a, b]) = [f (x m ), f (x M )].
25 Continuity in terms of inverse images of open sets There are even more ways to look at continuity. We know that f is continuous at p if ɛ > 0 δ > 0 x p < δ f (x) f (p) < ɛ, where δ is small enough to fit inside the circle B r (p) on which f is locally defined. Now suppose V was an open set in the range of f. This means every point in V can be written as y = f (x) for some point in the domain of f. Consider f 1 (V ), the inverse image of V under f. This is defined to be f 1 (V ) = {x dom(f ) : f (x) V } Is f 1 (V ) open? Let y 0 = f (x 0 ) V for some x 0 dom(f ). Since V is open, y 0 is an interior point of V and so there is an R > 0 so that B R (y 0 ) V. Then since f is continuous at x 0, choose ɛ = R and we see there is a δ > 0 so that if x x 0 < δ, where we choose δ < r, with f (x) f (x 0 ) < R. This says B δ (x 0 ) f 1 (V ) telling us x 0 is an interior point of f 1 (V ). Since x 0 is arbitrary, we see f 1 (V ) is open. Thus, the inverse image f 1 (V ) of an open set V is open.
26 Continuity in terms of inverse images of open sets We can use the idea of inverse images to rewrite continuity at a point p this way. ɛ > 0 δ > 0 x B δ (p) f (x) B ɛ (f (p)) or ɛ > 0 δ > 0 f (B δ (p)) B ɛ (f (p)) or ɛ > 0 δ > 0 B δ (p) f 1 (B ɛ (f (p))) Theorem f is continuous on the set D if and only if f 1 (V ) is an open set whenever V is open.
27 Continuity in terms of inverse images of open sets Proof ( ): We assume f is continuous on the set D. Let V in the range of f be an open set. By the arguments we have already shown you earlier, we know every point x 0 f 1 V is an interior point and so f 1 (V ) is an open set. ( ): We assume f 1 (V ) is an open set when V is open. Since V is open, given y 0 in V, there is a radius R so that B R (y 0 ) is contained in V. Choose any ɛ > 0. If ɛ R, argue this way: The set B R (y 0 ) is open and so by assumption, f 1 (B R (y 0 )) is open. Thus there is an x 0 f 1 (B R (y 0 )) so that y 0 = f (x 0 ). By assumption, x 0 must be an interior point of f 1 (B R (y 0 )). So there is a radius r so that B r (x 0 ) is contained in f 1 (B R (y 0 )). So x B r (x 0 ) f (x) B R (y 0 ). Said another way, x x 0 < r f (x) f (x 0 ) < R ɛ.
28 Continuity in terms of inverse images of open sets Proof On the other hand, if ɛ < R, we can argue almost the same way: instead of using the set B R (y 0 ), we just use the set B ɛ (y 0 ) which is still open. The rest of the argument goes through as expected! Comment Note here we are explicitly talking about continuity on the full domain not just at one point!
29 Continuity in terms of inverse images of open sets Proof On the other hand, if ɛ < R, we can argue almost the same way: instead of using the set B R (y 0 ), we just use the set B ɛ (y 0 ) which is still open. The rest of the argument goes through as expected! Comment Note here we are explicitly talking about continuity on the full domain not just at one point! This suggests how to generalize the idea of continuity to very general situations. A topology is a collection of subsets T of a set of objects X which is closed under arbitrary unions, closed under finite intersections and which contains both the empty set and the whole set X also. Our open sets in R as we have defined them are a collection like this. In this more general setting, we simply define the members of this set T to be what are called open sets even though they may not obey our usual definition of open.
30 Continuity in terms of inverse images of open sets Comment (Continued): Now the set X with the topology T is called a topological space and we denote it by the pair (X, T ). If (Y, S) is another topological space, based on what we have seen, an obvious way to generalize continuity is to say f : Ω X Y is continuous on Ω means f 1 (V ) is in T for all V S. Again, inverse images of open sets are open is a characterization of continuity. Note a set of objects can have many different topologies so continuity depends on the choice of topology really. For us we always use the topology of open sets in R which is nice and comfortable!
31 Continuity in terms of inverse images of open sets Comment (Continued): Now the set X with the topology T is called a topological space and we denote it by the pair (X, T ). If (Y, S) is another topological space, based on what we have seen, an obvious way to generalize continuity is to say f : Ω X Y is continuous on Ω means f 1 (V ) is in T for all V S. Again, inverse images of open sets are open is a characterization of continuity. Note a set of objects can have many different topologies so continuity depends on the choice of topology really. For us we always use the topology of open sets in R which is nice and comfortable! This is very abstract but it is generalizations like this that have let us solve some very hard problems!
32 Continuity in terms of inverse images of open sets Homework Prove lim x 1 14x + 25 exists. using an ɛ δ argument Prove lim x 2 (3x + 2)/(4x 2 + 8) exists using an ɛ δ argument f (x) = 3x 2 5 on [ 3, 6]. Find f ([ 3, 6]) which requires you to find the minimum and maximum of f on this domain. f (x) = 1/x 2 on (0, ). Find f (0, ). Note this is an interval. f (x) = 4x x on (0, 4). Is f (0, 4) an interval? f (x) = 5x 7 23x on ( 1, 6). Is f ( 1, 6) compact? This is meant to be a thought provoking question. In general, this is hard to answer. For example, if f (x) = sin(x), on ( π, π), we know the image f (( π, π)) is an interval. The domain is not compact, so we are NOT guaranteed that the image is compact. But here, the maximum occures at +1 and the minimum occurs at 1. So the range f (( π, π)) = [ 1, 1] which is compact even though the domain is not compact. So to answer this question, you need information about the minimum and maximum of this function. So graph it in MatLab, Sage etc and see what you find.
Consequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline Domains of Continuous Functions The Intermediate
More informationUpper and Lower Bounds
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 30, 2017 Outline 1 2 s 3 Basic Results 4 Homework Let S be a set of real numbers. We
More informationLower semicontinuous and Convex Functions
Lower semicontinuous and Convex Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2017 Outline Lower Semicontinuous Functions
More informationM17 MAT25-21 HOMEWORK 6
M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 Extremal Values 2
More informationThe Limit Inferior and Limit Superior of a Sequence
The Limit Inferior and Limit Superior of a Sequence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 13, 2018 Outline The Limit Inferior
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 2017 Outline Extremal Values The
More informationBolzano Weierstrass Theorems I
Bolzano Weierstrass Theorems I James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 8, 2017 Outline The Bolzano Weierstrass Theorem Extensions
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 Geometric Series
More information5.5 Deeper Properties of Continuous Functions
5.5. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 195 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested
More informationFilters in Analysis and Topology
Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2017 Outline Geometric Series The
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationThe Existence of the Riemann Integral
The Existence of the Riemann Integral James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 18, 2018 Outline The Darboux Integral Upper
More information2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α.
Chapter 2. Basic Topology. 2.3 Compact Sets. 2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. 2.32 Definition A subset
More informationUniform Convergence Examples
Uniform Convergence Examples James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 13, 2017 Outline 1 Example Let (x n ) be the sequence
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationMath 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.
Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations
More information6.2 Deeper Properties of Continuous Functions
6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 69 6.2 Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationConvergence of Sequences
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 2 Homework Definition Let (a n ) n k be a sequence of real numbers.
More informationProject One: C Bump functions
Project One: C Bump functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline 1 2 The Project Let s recall what the
More informationCOMPLEX ANALYSIS TOPIC XVI: SEQUENCES. 1. Topology of C
COMPLEX ANALYSIS TOPIC XVI: SEQUENCES PAUL L. BAILEY Abstract. We outline the development of sequences in C, starting with open and closed sets, and ending with the statement of the Bolzano-Weierstrauss
More information2 Metric Spaces Definitions Exotic Examples... 3
Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................
More informationUniform Convergence Examples
Uniform Convergence Examples James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 13, 2017 Outline More Uniform Convergence Examples Example
More informationMath 117: Continuity of Functions
Math 117: Continuity of Functions John Douglas Moore November 21, 2008 We finally get to the topic of ɛ δ proofs, which in some sense is the goal of the course. It may appear somewhat laborious to use
More informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
More informationMT804 Analysis Homework II
MT804 Analysis Homework II Eudoxus October 6, 2008 p. 135 4.5.1, 4.5.2 p. 136 4.5.3 part a only) p. 140 4.6.1 Exercise 4.5.1 Use the Intermediate Value Theorem to prove that every polynomial of with real
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationMath 410 Homework 6 Due Monday, October 26
Math 40 Homework 6 Due Monday, October 26. Let c be any constant and assume that lim s n = s and lim t n = t. Prove that: a) lim c s n = c s We talked about these in class: We want to show that for all
More informationConvergence of Sequences
Convergence of Sequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline Convergence of Sequences Definition Let
More informationReal Analysis. Joe Patten August 12, 2018
Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 23, 2018 Outline 1 Dirichlet
More information1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.
1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for
More informationWalker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015
Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of
More informationProofs Not Based On POMI
s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 1, 018 Outline Non POMI Based s Some Contradiction s Triangle
More informationConvergence of Fourier Series
MATH 454: Analysis Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April, 8 MATH 454: Analysis Two Outline The Cos Family MATH 454: Analysis
More informationU e = E (U\E) e E e + U\E e. (1.6)
12 1 Lebesgue Measure 1.2 Lebesgue Measure In Section 1.1 we defined the exterior Lebesgue measure of every subset of R d. Unfortunately, a major disadvantage of exterior measure is that it does not satisfy
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Mathematical Induction Simple POMI Examples
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 2, 207 Outline Mathematical Induction 2 Simple POMI Examples
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationUniform Convergence and Series of Functions
Uniform Convergence and Series of Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 7, 017 Outline Uniform Convergence Tests
More informationProofs Not Based On POMI
s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline 1 Non POMI Based s 2 Some Contradiction s 3
More informationMath 320-2: Final Exam Practice Solutions Northwestern University, Winter 2015
Math 30-: Final Exam Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A closed and bounded subset of C[0, 1] which is
More informationConstrained Optimization in Two Variables
in Two Variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 17, 216 Outline 1 2 What Does the Lagrange Multiplier Mean? Let
More informationA LITTLE REAL ANALYSIS AND TOPOLOGY
A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set
More informationDefinition: A "system" of equations is a set or collection of equations that you deal with all together at once.
System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationWriting proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases
Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationREAL VARIABLES: PROBLEM SET 1. = x limsup E k
REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationIntroduction to Topology
Chapter 2 Introduction to Topology In this chapter, we will use the tools we developed concerning sequences and series to study two other mathematical objects: sets and functions. For definitions of set
More informationGeneral Power Series
General Power Series James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 29, 2018 Outline Power Series Consequences With all these preliminaries
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationIntegration and Differentiation Limit Interchange Theorems
Integration and Differentiation Limit Interchange Theorems James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 11, 2018 Outline 1 A More
More informationThis chapter contains a very bare summary of some basic facts from topology.
Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the
More informationConstrained Optimization in Two Variables
Constrained Optimization in Two Variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 17, 216 Outline Constrained Optimization
More informationExam 2 extra practice problems
Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either
More informationIntroductory Analysis 1 Fall 2009 Homework 4 Solutions to Exercises 1 3
Introductory Analysis 1 Fall 2009 Homework 4 Solutions to Exercises 1 3 Note: This homework consists of a lot of very simple exercises, things you should do on your own. A minimum part of it will be due
More informationSequences. Limits of Sequences. Definition. A real-valued sequence s is any function s : N R.
Sequences Limits of Sequences. Definition. A real-valued sequence s is any function s : N R. Usually, instead of using the notation s(n), we write s n for the value of this function calculated at n. We
More informationMATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals
MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationThe Derivative of a Function
The Derivative of a Function James K Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 1, 2017 Outline A Basic Evolutionary Model The Next Generation
More informationClass Notes for MATH 255.
Class Notes for MATH 255. by S. W. Drury Copyright c 2006, by S. W. Drury. Contents 0 LimSup and LimInf Metric Spaces and Analysis in Several Variables 6. Metric Spaces........................... 6.2 Normed
More informationMATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE
MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE SEBASTIEN VASEY These notes describe the material for November 26, 2018 (while similar content is in Abbott s book, the presentation here is different).
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
More informationThe Banach-Tarski paradox
The Banach-Tarski paradox 1 Non-measurable sets In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the
More informationDifferentiating Series of Functions
Differentiating Series of Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 30, 017 Outline 1 Differentiating Series Differentiating
More informationMath 4603: Advanced Calculus I, Summer 2016 University of Minnesota Homework Schedule
Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Homework Schedule Notations I will use the symbols N, Z, Q and R to denote the set of all natural numbers, the set of all integers, the
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationBasic Definitions: Indexed Collections and Random Functions
Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery
More informationTHE INVERSE PROBLEM FOR DIRECTED CURRENT ELECTRICAL NETWORKS
THE INVERSE PROBLEM FOR DIRECTED CURRENT ELECTRICAL NETWORKS JOEL NISHIMURA Abstract. This paper investigates the inverse problem for the directed current networks defined by Orion Bawdon [1]. Using a
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More informationMTAEA Continuity and Limits of Functions
School of Economics, Australian National University February 1, 2010 Continuous Functions. A continuous function is one we can draw without taking our pen off the paper Definition. Let f be a real-valued
More information1 Homework. Recommended Reading:
Analysis MT43C Notes/Problems/Homework Recommended Reading: R. G. Bartle, D. R. Sherbert Introduction to real analysis, principal reference M. Spivak Calculus W. Rudin Principles of mathematical analysis
More information(x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11)
2.2 Limits and continuity In order to introduce the concepts of limit and continuity for functions of more than one variable we need first to generalise the concept of neighbourhood of a point from R to
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationMath 421, Homework #7 Solutions. We can then us the triangle inequality to find for k N that (x k + y k ) (L + M) = (x k L) + (y k M) x k L + y k M
Math 421, Homework #7 Solutions (1) Let {x k } and {y k } be convergent sequences in R n, and assume that lim k x k = L and that lim k y k = M. Prove directly from definition 9.1 (i.e. don t use Theorem
More informationDerivatives and the Product Rule
Derivatives and the Product Rule James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 Differentiability 2 Simple Derivatives
More informationLecture 2: A crash course in Real Analysis
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 2: A crash course in Real Analysis Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan Parthasarathy This lecture is
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline Dirichlet
More informationDR.RUPNATHJI( DR.RUPAK NATH )
Contents 1 Sets 1 2 The Real Numbers 9 3 Sequences 29 4 Series 59 5 Functions 81 6 Power Series 105 7 The elementary functions 111 Chapter 1 Sets It is very convenient to introduce some notation and terminology
More information2. The Concept of Convergence: Ultrafilters and Nets
2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two
More information5.5 Deeper Properties of Continuous Functions
200 CHAPTER 5. LIMIT AND CONTINUITY OF A FUNCTION 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested
More informationMath 421, Homework #9 Solutions
Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and
More informationThis exam contains 5 pages (including this cover page) and 4 questions. The total number of points is 100. Grade Table
MAT25-2 Summer Session 2 207 Practice Final August 24th, 207 Time Limit: Hour 40 Minutes Name: Instructor: Nathaniel Gallup This exam contains 5 pages (including this cover page) and 4 questions. The total
More information