Basic Analysis I — Real Analysis · Problem 3.4.10

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-continuity

Problem

a) Find a continuous f:(0,1)→Rf : (0, 1) \to \mathbb{R} and a sequence {xn}n=1∞\{x_n\}_{n=1}^\infty in (0,1)(0, 1) that is Cauchy, but such that {f(xn)}n=1∞\bigl\{f(x_n)\bigr\}_{n=1}^\infty is not Cauchy. b) Prove that if f:R→Rf : \mathbb{R} \to \mathbb{R} is continuous, and {xn}n=1∞\{x_n\}_{n=1}^\infty is Cauchy, then {f(xn)}n=1∞\bigl\{f(x_n)\bigr\}_{n=1}^\infty is Cauchy.

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