I did my research on this question and I found two answers that I feel contradict one another: this first one says it's not true and this second one says it is. When I asked my professor about it he said the following:
Let $f : (a,b) \rightarrow \mathbb{R}$ be a uniformly continuous function. Then $f$ can be continuously extended to $[a,b]$
Then if $f : [a,b] \rightarrow \mathbb{R}$ is a continuous function, then it is also uniformly continuous.
Therefore if $f : (a,b) \rightarrow \mathbb{R}$ is a uniformly continuous function, this implies $f : [a,b] \rightarrow \mathbb{R}$ is also uniformly continuous.
I'm inclined to believe that the second and my professor are correct, but the example of the first one is pretty compelling.
$f$ can be extended to a uniformly continuous function on $[a,b]$ but not every extension is uniformly continuous. In fact, not every extension is even continuous, as the example in that link shows.
There is a unique continuous extension and this extension is uniformly continuous.