Let {f}=1 is sequence of measurable functionsLet there exists E = {c: limf(x)}n-00Show that set E is measurable

Answered: Let {f} is sequence of measurable…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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a) f(a) exists b) f(x) is defined for 0<x<a c) f isn't continuous at x=a d)lim x-->a f(x) exists e)lim x-->a f(x)=f(a) which ones are true? could be up to four choices. explain.
Let f be a real valued function defined on all reals. Prove that f is continuous if and only if for each set O of real numbers, f^-1 (O) is an open set. (You may assume that fis continuous if and only if for each closed set C of real numbers, f^-1 (C) is a closed set)
Let f be a function defined for all x, and continuous at all values of x except at -2 and 0. Suppose that bx² + x for all x > 10 we have that b ≤ f(x) ≤2. Moreover, suppose that the following is known: lim_f(x) = 3, x 2 lim f(x) = f(-2) = 1, lim f(x) = f(0) = -b, x 2+ x-0 lim_ f(x) = 2. x-0+ (a) Does there always exist a value c on the interval (-3,0) such that f(c) = 0? Justify (b) Does there always exist a value c on the interval (-1,1) such that f(c) = 0? Justify (c) Evaluate the following limit: lim f(x). Justify In (a) and (b) we want to know if there always exists such a value c. Finding a specific example where such a c exists does not show that something always happens. The only thing you are allowed to assume about f(x) is the information given in the prompt.
Please teach me how to prove f is continuous by using the definition below. Let f:R →R be defined by f(x)= 2x-1. Definition: A function f:R→R is continuous if for each open subset V of R, f^-1(V) is an open subset of R.
Let f be a function defined for all x and continuous at all values of x except at x =-2 and x = 0. Suppose that for all x > 10, 4< f(x) < 4x2+x * and that the following limits are x2-4 known: lim f(x) = 3 x--2- lim f(x) = f(2) = 1 Xー-2+ lim f(x) = -4 lim f(x) = 2 Complete the following questions: a) Does there always exist a value c on the interval [-3, 0] such that f(c) = 0? Justify your answer. b) Does there always exist a value c on the interval [-1, 1] such that f(c) = 0? Justify your answer.
Plz do it in 10 min

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Let {f} is sequence of measurable functions Let there exists F = { x: limf, (x) E } 1-00 Show that set E is measurable