1549901369-Elements_of_Real_Analysis__Denlinger_

5.1 Continuity of a Tunction at a Point 231 (f) [_ is continuous at xo, if g(xo) #-0. g Proof. Exercise 14. • Theorem 5.1.14 (Co ...

232 Chapter 5 • Continuous Functions 0 < Ix - xol < 8' =? lf(x) - Yol < 8 =? lg (f(x)) - g (yo) I < c:. Therefore, l ...

5.1 Continuity of a Function at a Point 233 Thus (see Exercise 18) we conclude that The sine function is continuous everywhere. ...

234 Chapter 5 • Continuous Functions Sketch the graph of j(x) = Jx3 + 2x^2 + x on the interval [-2, 2]. Is f continuous at -1? ...

5.1 Continuity of a Function at a Point 235 16. Prove Corollary 5.1.15, Parts (a) and (b). [Hint: use Example 5.1.9 and Theorem ...

236 Chapter 5 • Continuous Functions Open Set Definition of Continuous: Prove that a function f : JR---+ JR is continuous every ...

5.2 Discontinuities and Monotone Functions 237 (c) Vr E Q, Vx E JR, f(rx) = [f(x)t. ( d) If f is continuous at 0, then it is con ...

238 Chapter 5 • Continuous Functions y 2 -1 -2 2 3 4 Figure 5.3 x Theorem 5.2.4 (Sequential Criterion for One-Sided Continuity) ...

5.2 Discontinuities and Monotone Functions 239 What is "removable" about a "removable discontinuity"? We shall see. Suppose f h ...

240 Chapter 5 • Continuous Functions Definition 5.2.12 (a) A function f is said to have an infinite discontinuity at xo if eithe ...

XI 5.2 Discontinuities and Monotone Functions 241 I I I I I I :JCxI 2) X2 x Increasing Figure 5.4 y I I I I I I I I I I :JCI x1) ...

242 Chapter 5 11 Continuous Functions Theorem 5.2.1 7 Suppose f is monotone increasing and bounded on an open interval I= (a, b) ...

5.2 Discontinuities and Monotone Functions 243 Corollary 5.2.19 If a function f is monotone on an interval I , then the only dis ...

244 Chapter 5 • Continuous Functions The characteristic function of a set A of real numbers defined by XA ( x) = {^1 ~f x E A ...

5.3 Continuity on Compact Sets and Intervals 245 Prove that Theorem 5.2.17 remains true if {f(x) : a < x < c} and {f(x) : ...

246 Chapter 5 • Continuous Functions Statement #1 is stronger than Statement #2; that is, #1 => #2 but #2 =fo #l. We give an ...

5.3 Continuity on Compact Sets and Intervals 247 That is , the function !IA is the same as the function f, except that its domai ...

248 Chapter 5 • Continuous Tunctions Part 2 ( {=): Suppose every sequence of points of A has a subsequence converging to a point ...

5.3 Continuity on Compact Sets and Intervals 249 Caution: Theorem 5.3.6 says that the continuous image of a closed, bounded set ...

250 Chapter 5 • Continuous Functions Case 1 (a< b): Since I is an interval, [a, b] ~I. Let A= {x E [a,b]: f(x) < w}. Then ...