Continuity and Uniform Continuity on half closed intervals I have been stuck on the following problem for a long time : 

Prove that if a function $f:(a,b]\to\mathbb R$ is continuous, then it is uniformly continuous if and only if $\lim_{x\to a^+}f(x)$ exists and is finite.


If $\lim\limits_{x\,\downarrow\,a} f(x)=y$, then the function that equals $y$ at $a$ and equals $f(x)$ at $x=$ anything besides $a$, is continuous on $[a,b]$; hence uniformly continuous.  A restriction of a uniformly continuous function to a subset of its domain is uniformly continuous.
