Review. April 12, Definition 1.2 (Closed Set). A set S is closed if it contains all of its limit points. S := S S


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1 Review April 12, Definitions nd Some Theorems 1.1 Topology Definition 1.1 (Limit Point). Let S R nd x R. Then x is limit point of S if, for ll ɛ > 0, V ɛ (x) = (x ɛ, x + ɛ) contins n element s S such tht s x. Tht is, (V ɛ (x)\{x}) S Ø. We let S = {x R x is limit point of S}. Definition 1.2 (Closed Set). A set S is closed if it contins ll of its limit points. Definition 1.3 (Closure of Set). Let S R. Then S := S S is the closure of S in R. Definition 1.4 (Open Set). A set U is open if the set R \U is closed. Theorem 1.5 (ɛ Criterion for Open Sets). A set U is open if nd only if, for ll x U, there exists ɛ x > 0 such tht (x ɛ x, x + ɛ x ) U. Definition 1.6 (Compctness). A set K R is compct if, given ny sequence ( n ) with n K, then ( n ) hs convergent subsequence ( nk ) such tht lim nk K. Theorem 1.7 (HeineBorel). A subset K R is compct if nd only if it is closed nd bounded. 1
2 1.2 Limits nd Continuity Definition 1.8 (Limit of Function). Let f : A R be function. Let R be limit point. A limit of f t is rel number l R such tht, for ll ɛ > 0, there exists δ > 0 such tht, if x A nd 0 < x < δ, then f(x) l < ɛ. If this is the cse, we write lim f(x) = l. x Theorem 1.9 (Sequence Criterion for Limits). Let f : A R be function. Suppose tht R is limit point of A. Then the following re equivlent: (i) lim x f(x) = l (ii) Given ny sequence (x n ) such tht x n A\{} nd lim n x n =, then lim n f(x n ) = l. Definition 1.10 (Continuity). Let A, B R, let f : A B nd A. We sy tht f is continuous t if, for every ɛ > 0, there exists δ > 0 such tht, if x A nd x < δ, then f(x) f() < ɛ. The function f is continuous if it is continuous t for every A. Theorem 1.11 (Imge of Compct Sets). Let f : K R be continuous function. Suppose tht K is compct. Then so is f(k). Theorem 1.1 (Extreme Vlue Theorem). Let f : K R be continuous function on nonempty compct set K. Then f(k) hs mximum nd minimum. Tht is, there exists x 0 nd x 1 in K such tht, for ll x K, f(x 0 ) f(x) f(x 1 ). Theorem 1.12 (Intermedite Vlue Theorem). Let f : [, b] R be continuous function. If r R is such tht f() < r < f(b) or f(b) < r < f(), then there exists c (, b) such tht f(c) = r. Definition 1.13 (Uniform Continuity). A function f : A R is uniformly continuous if for ll ɛ > 0, there exists δ > 0 such tht for ll x nd y in A, if x y < δ, then f(x) f(y) < ɛ. Theorem 1.14 (Uniform Continuity on Compct Sets). Let f : K R be continuous function. If K is compct set, then f is uniformly continuous. 2
3 1.3 Differentition Definition 1.15 (Derivtive). Let A be n open set nd let f : A R. Let A. Then, if it exists, the limit f () = lim x f(x) f() x is clled the derivtive of f t nd we sy tht f is differentible t. If f is differentible t for every A, then f is differentible on A. Theorem 1.16 (Chin Rule). Let A, B R be open sets. Let f : A R nd g : B R nd suppose tht f(a) B. Let A. Suppose tht f is differentible t nd g is differentible t f(). Then g f : A R is differentible t nd (g f) () = g (f()) f (). Definition 1.17 (Mxim nd Minim). Let f : A R. f ttins its mximum t A if f() f(x) for ll x A. f ttins its minimum f() f(x) for ll x A. f ttins its locl mximum t A if there is region (b, c) A such tht f() f(x) for ll x (b, c). f ttins its locl minimum t A if there is region (b, c) A such tht f() f(x) for ll x (b, c). Theorem 1.18 (Vnishing of Derivtive t Mxim nd Minim). Suppose tht f : A R is differentible on A (with A open). Suppose tht f ttins locl mximum or locl minumum t A. Then f () = 0. Theorem 1.19 (Rolle s Theorem). Let f : [, b] R be continuous. Suppose tht f is differentible on (, b). If f() = f(b) = 0, then there exists c (, b) such tht f (c) = 0 Theorem 1.20 (Men Vlue Theorem). Let f : [, b] R be continuous. differentible on (, b). Then there exists c (, b) such tht Suppose tht f is f(b) f() = f (c)(b ). Definition 1.21 (Incresing nd Decresing). A function f : A R is 3
4 incresing if, for ll, b A with < b, f() f(b). It is strictly incresing if, for ll, b A with < b,f() < f(b). decresing if, for ll, b A with < b, f() f(b)., b A with < b,f() > f(b). It is strictly decresing if, for ll Theorem 1.22 (Sign of derivtive). Let f : [, b] R be continuous. Suppose tht f is differentible on (, b). () If f (x) > 0 for ll x (, b), then f is strictly incresing on [, b]. (b) If f (x) < 0 for ll x (, b), then f is strictly decresing on [, b]. (c) If f (x) = 0 for ll x (, b), then f is constnt [, b]. 1.4 Integrtion Definition 1.23 (Prtitions). A prtition of [, b] is finite subset P of [, b] which contins the points nd b. Definition 1.24 (Lower nd Upper Sums). Let f : [, b] R be bounded function. P = {t 0, t 1,..., t n } be prtition of [, b]. Let Let m i = inf{f(x) t i 1 x t i } M i = sup{f(x) t i 1 x t i }. The lower sum of f for P is L(f, P ) = The upper sum of f for P is U(f, P ) = n m i (t i t i 1 ). i=1 n M i (t i t i 1 ). i=1 Definition 1.25 (Upper nd Lower Integrl). Let f : [, b] R be bounded function. Then L(f) = sup{l(f, P ) P is prtition of [, b]} is clled the lower integrl of f from to b nd U(f) = inf{u(f, P ) P is prtition of [, b]} 4
5 is clled the upper integrl of f from to b. Definition 1.26 (Integrbility). Let f : [, b] R be bounded function. If L(f) = U(f), then f is integrble on [, b]. The vlue L(f) = U(f) is clled the integrl of f from to b. It is denoted Theorem 1.27 (ɛ Criterion for Integrbility). Let f : [, b] R be bounded function. Then f is integrble if nd only if, for every ɛ > 0, there exists prtition P ɛ of [, b] such tht f. U(f, P ɛ ) L(f, P ɛ ) < ɛ. Theorem 1.28 (The Fundmentl Theorem of Clculus). Let f be bounded function which is integrble on [, b]. 1. Suppose tht f = F on (, b) for continuous function F : [, b] R which is differentible on (, b). Then 2. Let F : [, b] R be given by f = F (b) F (). F (x) = x If f is continuous t p (, b), then F is differentible t p nd f. F (p) = f(p). Theorem 1.29 (Integrtion by Prts). Let f, g : A R for n open set A contining [, b], nd such tht f nd g re continuous on [, b]. Then fg = (f(b)g(b) f()g()) Theorem 1.30 (Chnge of Vribles). Let g : A R where A is n open set contining [, b]. Let f : [c, d] R nd suppose tht g([, b]) [c, d]. Suppose tht g is continuous on [, b] nd f is continuous. Then, (f g) g = g(b) g() f. f g. 5
6 2 Some Prctice Problems (1) Let K 1, K 2 R be compct sets. Show tht K 1 K 2 is compct. (2) Suppose tht x is limit point of S nd U is n open set contining x. Show tht x is limit point of S U. (3) Prove tht if f : R R nd g : R R re continuous, then so is the composite g f. (4) Let X be finite subset of R. Prove tht ny function f : X R is continuous. (5) Suppose tht f : [, b] R is continuous nd tht it is differentible on (, b). (i) Suppose tht f hs mximum t c (, b). Prove tht f (c) = 0. (ii) Suppose tht f (x) > 0 for ll (, b). Prove tht f is incresing. (6) Let f : [, b] R be bounded incresing function. Prove tht f is integrble. (7) Suppose tht f : [, b] R is continuous, nd for every [c, d] [, b], suppose tht d c f = 0. Prove tht f(x) = 0 for ll x [, b]. 6
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