CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS

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1 CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS DEFINITION. Let a R and let I be an open interval contains a, and let f be a real function defined everywhere except possibly at a. Then f(x) is said to converge to L, as x approaches a, if and only if for every ǫ > 0 there is a δ(ǫ) such that 0 < x a < δ(ǫ) implies f(x) L < ǫ. In this case we write lim x a f(x) = L and call L the limit of f(x) as x approaches a. Example. f(x) = mx + b. Example. f(x) = x 2 + x 3. Remark. Let a R, let I be an interval that contains a, and let f, g be real functions defined everywhere on I except possibly at a. If f(x) = g(x) for all x I \ {a} and lim x a f(x) = L, then lim x a g(x) = L. Example. g(x) = x3 +x 2 x 1 x 1, f(x) = x + 1. Theorem. [Sequential Characterization of Limits] Suppose that a R, let I be an open interval contains a, and let f be a real function defined everywhere on I except possibly at a. Then lim x a f(x) = L if and only if f(x n ) L for every sequence {x n } I \ {a} that converges to a as n. Example. f(x) = sin 1 x if x 0 and f(0) = 0. Theorem. Suppose that a R, that I is an open interval that contains a, and that f, g are real functions defined everywhere on I except possibly at a. If f(x) and g(x) converge as x approaches to a, then so do (f +g)(x), (fg)(x), (cf)(x), and (f/g)(x) (when the limit of g(x) is nonzero). In fact and (when the limit of g(x) is nonzero) lim(f + g)(x) = lim f(x) + lim g(x), x a x a x a lim(fg)(x) = (lim f(x))(lim g(x)), x a x a x a lim(cf)(x) = c(lim f(x)), x a x a lim (f x a g )(x) = lim x a f(x) lim x a g(x). Typeset by AMS-TEX 1

2 2 Theorem. [Squeeze Theorem for Function] Suppose that a R, let I be an open interval contains a, and and that f, g, h are real functions. defined everywhere on I except possibly at a. (1) If g(x) h(x) f(x) for all x I \ {a} and lim x a f(x) = lim x a g(x) = L, then lim x a h(x) = L. (2) If g(x) < M for all x I \ {a} and lim x a f(x) = 0, then lim x a f(x)g(x) = 0. Theorem. [Comparison Theorem for Functions] Suppose that a R, that I is an open interval that contains a, and that f, g are real functions defined everywhere on I except possibly at a. If f(x) and g(x) converge as x approaches to a, and f(x) g(x) for all x I \ {a}, then lim x a f(x) lim x a g(x). Example. lim x 1 x 1 3x ONE SIDED LIMITS AND LIMITS AT INFINTY DEFINITION. Let a R. (1) A real function f is said to converge to L as x approaches a from the right if and only if f is defined on some open interval I with left endpoint a and for every ǫ > 0 there is a δ(ǫ) such that a + δ(ǫ) I and a < x < a + δ(ǫ) implies f(x) L < ǫ. In this case we call that L is the righthand limit of f at a and denote it by f(a+) := L =: lim x a+ f(x). (2) A real function f is said to converge to L as x approaches a from the left if and only if f is defined on some open interval I with right endpoint a and for every ǫ > 0 there is a δ(ǫ) such that a δ(ǫ) I and a > x > a δ(ǫ) implies f(x) L < ǫ. In this case we call that L is the lefthand limit of f at a and denote it by f(a ) := L =: lim x a f(x). Example. f(x) = x + 1 if x > 0 and f(x) = x 1 if x 0. Example. lim x 0+ x = 0.

3 3 Theorem. Let f be a real function, then lim x a f(x) = L if and only if lim x a+ f(x) = L = lim x a f(x). DEFINITION. We said that f(x) L as x + (respectively, as x ) if and only if there exists a c > 0 such that (c, + ) Dom(f) (respectively, (, c) Dom(f)) and given ǫ > 0 there is M R, such that x > M (respectively, x < M) implies f(x) L < ǫ. In this case we write lim f(x) = L (respectively, lim x f(x) = L.) x DEFINITION. Let a R and let I be an open interval contains a, and let f be a real function defined everywhere except possibly at a. Then f(x) is said to converge to + (respectively, ), as x approaches a, if and only if for every M R there is a δ(m) such that 0 < x a < δ(m) implies In this case we write Example. lim x 1 x = 0. Example. lim x 1 x+2 2x 2 3x+1 =. UNIFY NOTATION f(x) > M (respectively, f(x) < M) lim f(x) = + (respectively, lim f(x) =. x a x a lim f(x). x a,x I Theorem. Let a be an extended real number and I be a nondegenerated open interval that either contains a or has a as one of its endpoint. Suppose further that f is a real function defined on I except possibly at a. Then lim x a,x I f(x) exists and equals to L if and only if f(x n ) L for all sequences x n I that satisfy x n a and x n a as n. Example. lim x 2x x 2 = CONTINUITY DEFINITION. Let E be a nonempty subset of R and f : E R. (1) f is said to be continuous at a point a E if and only if given ǫ > 0 there is a δ > 0 such that x a < δ and x E implies f(x) f(a) < ǫ. (2) f is said to be continuous on E if and only if f is continuous at every x E.

4 4 Theorem. Suppose that E is a nonempty subset of R, A E, and f : E R. Then the following statements are equivalent: (1) f is continuous at a. (2) If x n converges to a and x n E, then f(x n ) f(a) as n. Theorem. Let E be a nonempty subset of R and f, g : E R. If f, g are continuous at a point a E (respectively, continuous on the set E), then so are f + g, fg, cf(for any c R). Moreover, f/g is continuous at a when g(a) 0 (respectively, on E when g(x) 0 for all x E). Remark. If f, g are continuous, so are f, f +, f, f g, f g. DEFINITION. Suppose that A and B are subsets of R and that f : A R and g : B R. If f(a) B, then the composition of g with f is the function g f : A R defined by g f(x) = g(f(x)), x A. Theorem. Suppose that A and B are subsets of R and that f : A R and g : B R, and f(a) B. (1) If A = I \ {a}, where I is a nondegenerated interval that contains a or has a as one of its endpoint, if L = lim x a,x I f(x) exists and belongs to B, and if g is conuous at L B, then lim g f(x) = g( lim f(x)). x a,x I x a,x I (2) If f is continuous at a and g is continuous at f(a) B, then g f is continuous at a A. DEFINITION. Let E be a nonempty subset of R. A function f : E R is said to be bounded on E if and only if there is a M R such that f(x) < M for every x E. Theorem. [EXTREME VALUE THEOREM]. If I is closed bounded interval and f : I R is continuous on I, then f is bounded on I. Moreover if M = sup f(i) and m = inf f(i), then there exist x M, x m I such that f(x M ) = M and f(x m ) = m. Remark. The extreme value theorem is false if either closed or bounded is dropped from the hypotheses.

5 5 Lemma. [Sign-Preserving property] Let f : I R where I is a nondegenerated open interval. If f is continuous at a point a I and f(a) > 0, then there are positive numbers ǫ, δ such that x a < δ implies f(x) > ǫ. Theorem. [Intermediate Value Thoerem] Let I be a nondegenerated open interval and f : I R be continuous. If a, b I with a < b, and if y 0 lies between f(a) and f(b), then there is an c (a, b) such that f(c) = y 0. Example. f(x) = x x, x 0 and f(0) = 1. Example. f(x) = sin 1 x, x 0 and f(0) = 1. Example. [Dirichlet Function] f(x) = 1 if x Q and f(x) = 0 if x / Q. Example. f(x) = 1 q if x = p q (in reduced form), and f(x) = o, x / Q. Example. g(x) = 1, x 0 and g(0) = 0, then g f is the Dirichlet function. 3.4 UNIFORM CONTINUITY DEFINITION. Let E be a nonempty subset of R and f : E R. Then f is said to be uniformly continuous on E (notation f : E R is uniformly continuous) if and only if for any ǫ > 0 there is a δ(ǫ) such that x a < δ and a, x E imply f(x) f(a) < ǫ. Example. f(x) = x 2 on (0, 1) and on R. Lemma. Suppose that E R and f : E R is uniformly continuous. If x n E is Cauchy, then f(x n ) is Cauchy. Theorem. Suppose that I is a closed,bounded interval. If : I R is continuous on I, then f is uniformly continuous on I. Theorem. Let (a, b) be a bounded, open, nonempty interval and f : (a, b) R. Then f is uniformly continuous on (a, b) if and only id f can be extended continuously to [a, b], i.e. if and only if there is a continuous function g : [a, b] R that satisfies g(x) = f(x) for all x (a, b). Example. f(x) = (x 1) log x on (0, 1).