"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 [tex]f(x)=\left\{\begin{array}{cc}x &\mbox{ if }x\geq 1\\ \frac{1}{x-1} &\mbox{ if }x<1\end{array}\right[/tex]
Hah! The question forgot to state that f is supposed to be a continuous map [itex][0, +\infty) \rightarrow \mathbb{R}[/itex]!