# Show that $f$ is uniformly continuous.

Suppose that $f:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $\lim\limits_{x \rightarrow a}f(x)$ exists. Show that $f$ is uniformly continuous.

I am really struggling with this one. HELP

• You can start by remembering that $F$ is uniformly continuous on $[a+\epsilon,b]$ for any $\epsilon >0$ (because $F$ is continuous on that interval, which is closed). Now, what you have to do, is try to show that if you pick $\epsilon$ small enough, $F$ is also bounded on $(a,a+\epsilon]$ for the same bound given above. This is where you use that $\lim_{x\rightarrow a} f(x)$ exists. – James Apr 20 '15 at 17:32
• You can easily extend $f$ to a function from $[a,b]$ to $\mathbb R$. – Akiva Weinberger May 12 '15 at 11:23

With this in mind and with the existence of the limit you get that your function can be extended to $[a,b]$, which is compact. Since the extension is uniformly continuous the restriction to $(a,b]$ is uniformly continuous as well.