Real analysis 6. Let E1, E2, E3, ... be subsets of R and suppose that f is defined and is uniformly continuous on eachof these sets. Prove that for every natural number n, ƒ is uniformly continuous on the unionU Ek-k=1However, give a counterexample showing that f need not be uniformly continuous on the unionU Ek-k=1

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6. Let E1, E2, E3, ... be subsets of R and suppose that f is defined and is uniformly continuous on each of these sets. Prove that for every natural number n, f is uniformly continuous on the union U Ek- k=1 However, give a counterexample showing that f need not be uniformly continuous on the union U Ek- k=1

6. Let E1, E2, E3, ... be subsets of R and suppose that f is defined and is uniformly continuous on each of these sets. Prove that for every natural number n, f is uniformly continuous on the union U Ek- k=1 However, give a counterexample showing that f need not be uniformly continuous on the union U Ek- k=1

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 19E

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Real analysis

6. Let E1, E2, E3, ... be subsets of R and suppose that f is defined and is uniformly continuous on each
of these sets. Prove that for every natural number n, ƒ is uniformly continuous on the union
U Ek-
k=1
However, give a counterexample showing that f need not be uniformly continuous on the union
U Ek-
k=1

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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