# Uniform Continuity

by Icebreaker
Tags: continuity, uniform
 P: n/a "Suppose f:[0, inf) -> R is such that f is uniformly continuous on [a, inf) for some a>0. Prove that f is uniformly continuous on [0, inf)." But this is not true, is it? Consider the function $$f(x)=\left\{\begin{array}{cc}x &\mbox{ if }x\geq 1\\ \frac{1}{x-1} &\mbox{ if }x<1\end{array}\right$$
 Emeritus Sci Advisor PF Gold P: 16,091 Hah! The question forgot to state that f is supposed to be a continuous map $[0, +\infty) \rightarrow \mathbb{R}$!