Basic Analysis I — Real Analysis · Problem 3.4.9

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-continuity

Problem

(Challenging) Let

f : \mathbb{Q} \to \mathbb{R}

be a uniformly continuous function. Show that there exists a uniformly continuous function

\widetilde{f} : \mathbb{R} \to \mathbb{R}

such that

f(x) = \widetilde{f}(x)

for all

x \in \mathbb{Q}