Classifing functions with the property that the preimage of path-connected sets is path-connected

Question

Given two path-connected topological spaces $\mathcal{X},\mathcal{Y}$, consider a function $f: \mathcal{X} \to \mathcal{Y}$ with the property, that the preimage $f^{-1}(V)$ is pathconnected for all path-connected $V \subset \mathcal{Y}$. For example, all homeomorphisms and constant functions should fall into this class. Does anybody know a name for this class of functions fulfilling this or a similar condition? I guess for functions from $\mathbb{R} \to \mathbb{R}$, the condition mentioned above should coincide with monotonicity, and I was wondering about a more general case for other domains and codomains.