The other day in a exam, I was given the following exercise:
Given $f : [0,1] \to \mathbb{R}$ continuous and such that $f(0) = 0, f(1) = 1$, let $g : \mathbb{R} \to \mathbb{R}$ be $g(x) = [x] + f(x - [x])$. Prove that $g$ is uniformly continuous.
I'm looking for more examples of this kind of exercise to practice with (i.e. functions with uniform continuity that are not as straightforward to prove that they are).
Any continuous function $f$ from a closed interval $[a, b]$ to $[a, b]$ is uniformly continuous. See https://en.wikipedia.org/wiki/Heine–Cantor_theorem.
If $f(a) = a$ and $f(b) = b$ then extending $f$ to a function $\Bbb{R} \to \Bbb{R}$ by taking $f(x) = x$ for $x \not\in [a, b]$ still gives a uniformly continuous function.
If you want practice on proving uniform continuity from first principles, you can create your own examples from this.