Locally one-to-one mappings and a classical theorem on schlicht functions

Some topological theorems are given which are similar to Darboux's Theorem and from one of which (Theorem 1) Darboux's Theorem is easily deduced. Our theorems are like Darboux's Theorem in the sense that we assume a certain mapping to be one-to-one on the boundary of its domain and conclude that it is one-to-one everywhere on its domain. (Author)