Is every uniformly continuous function 1-1 and onto?

Question

Let $f : (X,d)\rightarrow (Y,\rho)$. Is $f$ 1-1 and onto if $f$ is a uniformly continuous function on X?
If not, would $X$ being compact change things?
If not, do you know a theorem or something similar to this?

Thanks in advance.

Answer 1

For a counterexample for both, consider a constant function when the domain and codomain both have more than one element.