Orthosymmetrical monotone functions

Abstract

A straightforward generalization of the classical inverse of a real function based on reflections leads to several insuperable difficulties. We introduce a new type of inverse w.r.t. monotone bijections $\phi$ that is determined by the direction of the base vectors of the real Euclidean plane. Inverting a monotone function in the real plane does not necessarily result in a function. Given an increasing real function $f$, Schweizer and Sklar geometrically construct a set of inverse functions. We will largely extend their construction to our new concept of $\phi$-inverses, also incorporating decreasing functions $f$. Furthermore, the geometrical and algebraical aspects of our approach are elaborated comprehensively. Special attention goes to the symmetry of a monotone function $f$ w.r.t. some monotone bijection $\phi$.