I'm trying to find an example of a uniformly continuous function $f:X\to Y$ such that $f(X)$ is not bounded, for some bounded metric space $X$ and a metric space $Y$.
The classic exercise is to show that if $X\subseteq\mathbb{R}$ is bounded then any uniformly continuous $f$ sends bounded sets onto bounded sets, this is because such sets are totally bounded in this case.
So what is an example of a bounded metric space $X$, not totally bounded, such that a uniformly continuous function sends it to a not bounded set?
