Connected and connectivity functions on dendrites (Q1124156)

For real-valued functions (with connected domains) the properties of being connected (i.e. having a connected graph) and of being a connectivity function (i.e. being connected if it is restricted to connected subsets of the domain) are considered. Clearly, connectivity implies connectedness. There is disproved a conjecture in the opposite direction, namely that connected functions defined on connected subsets of dendrites are necessarily connectivity functions. For dendrites with only finitely many branch points the conjecture is confirmed.