If $f:X \rightarrow Y$ is uniformly continuous and $X$ is complete then so is $f(X)$?
My idea is that this is not true. $f :\mathbb{R} \rightarrow (-\pi/2,\pi/2)$, by $f(x)=tan^{-1}(x)$.
I think this result is true only if $f(X)=Y$.
Can anyone suggest some comments on this idea?