I'm interested in continuous maps between topological spaces $f:X\to Y$ such that for any compact subset $L$ of $Y$ contained in $f(X)$, there is a compact subset $K$ of $X$ such that $L$ is contained in $f(K)$.

Proper maps satisfy this, but there are examples of continuous maps which don't, for example with discrete spaces, taking a non-stationary convergent sequence extended at infinity.

I would like to know if there are characterizations for those topological spaces which have enough compact subsets in the sense that: each real-valued continuous function satisfy this property.