Uniform Continuity

1. Icebreaker

0
"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

$$f(x)=\left\{\begin{array}{cc}x &\mbox{ if }x\geq 1\\ \frac{1}{x-1} &\mbox{ if }x<1\end{array}\right$$

2. Hurkyl

16,089
Staff Emeritus
Hah! The question forgot to state that f is supposed to be a continuous map $[0, +\infty) \rightarrow \mathbb{R}$!

3. Icebreaker

0
Phew! It's not just me then.