Question

real analysis

Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is not continuous on the interval I, or i Exercises 16, 17, and 18.) 160 the interval Cocn o. EXERCISES 4.4 For each of the following functions / defined on R\ (o), find lim, /x) and lim, ) provided the limits [x] 0, 1. Prove Theorem 4.4.3 ske sh 3. For each of the functions f in Exercise 2, determine whether f has a removable ity, or a discontinuity of second kind, at 0. If f has a removable discontinuity be defined so that f is continuous at 0. at 0, specify how f(0) shold Investigate continuity of g (x)2) [x] at2. Let fx) x [x]. Discuss continuity of f. Sketch the graph of f. sLet/ erlx), X uch that unifor n y continuo sepe , evi fbe defined by f(a) 18 x>-2. x+b, Determine the value of b such that f has a removable discontinuity atx2. メ*Prove that g(x) sin(2m[x)) is not uniformly continuous on R. 8· Prove that the function f of Example 4.4.1 1 (a) is continuous at x-1. チLet E R and let f be a real-valued function on E. Suppose p E R is a limit point of E n (p, oo). Prove that lim.)L if and only if lim f(P) L for every sequence (pa) in E with p, > p for all n E N and p. I Let f be a real-valued function on (a, b a. If f is continuous on (a, b] and lim f(x) exists, prove that f is uniformly continuous on (a, b] b. If f is uniformly continuous on (a, b], prove that lim f(x) exists. Let f: [0, 2] → R be defined by Show that f andare strictly increasing and find f([0, 2]). Are f and f 1 continuous at every point of their respective domains? *If m E Z, n E N, prove that f(x) = xmm s continuous on (0,0)

1,3,8,11,12 please

4.4.3

4.4.11a

Answer 1

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