Arcwise increasing maps

A surjective continuous map f:[0,1]→Xf:[0,1]→X is called an arcwise increasing map if for any two closed subintervals A and B   of [0,1][0,1] such that A⊊BA⊊B, then f(A)⊊f(B)f(A)⊊f(B). A continuum X is said to admit an arcwise increasing map if there is an arcwise increasing map onto X. It is shown that any Peano continuum with no free arcs admits an arcwise increasing map, and a characterization of graphs and dendrites that admit arcwise increasing maps is given.