TITLE: Does there exist an unbounded function that is uniformly continuous?
QUESTION [8 upvotes]: I know that $1/x$ is unbounded on $(0,5)$ (for example) and that since it is unbounded, it is not uniformly continuous. 
Does a function have to be bounded to be uniformly continuous? I don't think one exists.

REPLY [2 votes]: Just to add that a bounded derivative is sufficient for an (obviously differentiable) function $ f:\mathbb R \rightarrow \mathbb R$ to be uniformly-continuous, but a necessary condition for the limit as $ x\rightarrow \infty $ to be finite (when the limit $f'(x)$ exists as $x \rightarrow \infty$) is that lim$_{x\rightarrow \infty}f'(x)=0. $ So, to make your statement true, you need to add these two conditions: the limit of $f'(x)$ exists as you go to $\infty$ , and it is $0$.
EDIT: As pointed out by timur, these conditions are necessary, but not sufficient, i.e., we need $f'(x) \rightarrow 0$ for the limit to be finite, a.k.a., for f to be bounded . I suspect that adding a condition on $|f''(x)| \rightarrow \infty$ is sufficient, i.e., the function decreases fast-enough to not become unbounded.