In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ were sent by $f$ to sequences lying in a compact set in $Y$ (in particular, any subsequence of $f(x_n)$ had a convergent subsequence).

Do these kind of maps have a name ?

Thanks.