TITLE: Function $\mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but not uniformly continuous
QUESTION [20 upvotes]: I found an example of a function $f: \mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but is not uniformly continuous. It is $\sin(x^2)$. I think it's not uniformly continuous because the derivative is bigger and bigger as $x$ increases. But I don't know how to prove this is uniformly continuous. Is $\sin(x^2)$ uniformly continuous then? if it isn't, can you guys think of any other examples? thanks

REPLY [2 votes]: By MVT $|\sin(x^2)-\sin(y^2)|=|2k \cos(k^2)||x-y|$,where $k$ is between $x$ and $y$ which is not always less than $|x-y|$, for all $x,y \in \mathbb{R}$. Thus $\sin(x^2)$ is not uniformly continuous on $\mathbb{R}$.