1549901369-Elements_of_Real_Analysis__Denlinger_

5.3 Continuity on Compact Sets and Intervals 251 For all n E N, we define tn = min { c + ~, b}. Thus, 'Vn E N, c < tn :::; c ...

252 Chapter 5 11 Continuous Functions Proof. Exercise 11. • The intermediate value theorem furnishes one of the principal tools ...

5.3 Continuity on Compact Sets and Intervals 253 Proof. Suppose a :::; b, and f : [a, b] ___, [a, b] is continuous. Define the f ...

254 Chapter 5 • Continuous Functions Proof. If n = 1 we merely take y = xo. Suppose n ~ 2 in N, and x 0 > 0 in R Consider the ...

5.3 Continuity on Compact Sets and Intervals 255 Prove by example that (a) the continuous image of a bounded set need not be b ...

256 Chapter 5 11 Continuous Functions Prove that if f : I--+ JR. is continuous on an interval I , and f (I) contains only ratio ...

5.4 Uniform Continuity 257 We now consider a little more closely what it means for a function to be contin- uous on a set. We sh ...

258 Chapter 5 • Continuous Functions Solution. Let f ( x) = 3x^2 - 2x - l. This is the same function used in Example 5.1.2. You ...

5.4 Uniform Continuity 259 N otes: (1) The conclusion of Theorem 5.4.3 cannot be strengthened to say that f : V(f) -->IR is c ...

260 Chapter 5 • Continuous Functions Lemma 5.4.5 (Negation of Uniform Continuity) A function f : D(f) ---+ JR is not uniformly c ...

5.4 Uniform Continuity 261 For each i, /xi-Xi-1/ = Xi-Xi-1 = [a+ ib~a] - [a+ (i - l)b~a] = b~a < o. To see that f is bounded ...

262 Chapter 5 • Continuous Functions Since A is closed, Theorem 3.2.19 says that L E A. Thus, f is continuous at L. Hence, by th ...

5.4 Uniform Continuity 263 Proof. Part 1 (=>): See Theorem 5.4.8 above. Part 2 ( <=): We shall prove the contrapositive. S ...

264 Chapter 5 • Continuous Functions *Definition 5.4.11 Suppose f : V(f) . JR and V(f) ~ A. Then a function g : A. JR is said to ...

5.4 Uniform Continuity 265 *corollary 5.4.13 A continuous f: (a,b)-+ ~is uniformly continuous on (a, b) {::} both lim f(x) and l ...

266 Chapter 5 • Continuous Functions Prove that the function f(x) = .jX is uniformly continuous on [O, + oo). [Hint: first prov ...

5.5 *Monotonicity, Continuity, and Inverses 267 (b) Prove that the conjecture is true if A and B are bounded and sup A < inf ...

268 Chapter 5 • Continuous Functions Since f is monotone increasing on I, and f(xi) < f(xo) < f(x2), we must have x1 < ...

5.5 *Monotonicity, Continuity, and Inverses 269 Proof. We prove the strictly increasing case, and leave the strictly decreas- in ...

270 Chapter 5 • Continuous Functions while and c < x < d ::::} f(x) lies between f(c) and f(d) * f(d) < f(x) < f(c) ...