real analysis - Does a function have to be bounded to be uniformly continuous?

My book defines uniform continuity as a form of continuity that works for any points $a$ and $x$ in an interval $I$ such that

$$|x-a| < \delta$$ implies that $$f(x) - f(a) < \epsilon$$

It then goes on to assert that "If $f$ is continuous over a closed and bounded interval $[a,b]$, it is uniformly continuous on said interval."

My question is this: Does $f$ have to be bounded to be uniformly continuous? If not, can someone give me an example and show me why this is the case? This is a concept that I've only been shown with bounded examples in class (and we don't have class until after Thanksgiving).

I saw there exists a question here like this, but I didn't feel the answer was rigorous enough for me to understand fully.