I'm writing some lecture notes on Riemann/Darboux integration and I recently finished the proof that continuous implies that the Riemann integral exists. The proof of the statement relies on showing that continuous functions defined in an interval are uniformly continuous.
As i finished doing that part, I started wondering: what are all the continuous functions that are not uniformly continuous. I know that functions with fast growing and with fast oscilation are in this category ($f(x) = e^x$ or $f(x) = \sin{(e^x)}$). Is there any other example (not explicit example, i don't care about what are all the functions, I want all the classes of functions that have a similar look and that have some property that make them non-uniformly continuous)?
$f(x)=x^2$ might be the easiest example of a continuous function which is uniformly continuous. Similarly $f(x)=x^a$ with $a>1$.
More generally if $f$ is differentiable and $$\lim_{x\rightarrow\infty} |f'(x)|=\infty$$ then $f$ is not uniformly continuous on any interval $[a,\infty[$. This does formalize your notions of "fast growing" and "fast oscillation".