About linear preimages of continuous maps, that preserve orientation of triangles

The article describes the differential properties of continuous mappings $f:D\to R^n$, which retain the orientation of some simplexes in advance of this subset of $S(D)$. Such mappings represent a natural generalization of the class of monotone functions of one variable. In this paper we prove that the mapping monotonic in this sense have to be affine. In addition, we prove a generalization of this result, provided that the map preserves the orientation of an open family of simplexes. As a consequence, we obtain a result on the structure of the inverse image of a straight monotone mapping of plane. Namely, the main result is Theorem.
Тheorem Let $f:D\to R^2$ be mapping preserves the orientation of triangles with obtuse angle $\gamma, \pi/2<\alpha<\gamma<\beta<\pi$. Then if the inverse image of a straight line $L$ is nowhere dense, then $L$ is union of a finite or countable number of locally Lipschitz curves.