Show that $f$ is uniformly continuous.

Question

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

Answer 1

Do you already know the fact that continuous functions from a compact set are uniformly continuous? (This follows from the fact that continous functions coming from compact sets always have a maximum)

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.

Answer 2

"A continuous function on a closed set is always Uniform"

if it is not closed then just check whether the limit exists at the end points, if it exists then the function is uniformly continuous otherwise not.