Orthosymmetrical monotone functions.(English)Zbl 1142.26007

The authors introduce a new type of inverse of a set $$F\subseteq \mathbb{R} ^{2}$$ with respect to a monotone $$\mathbb{R}\rightarrow \mathbb{R}$$ bijection $$\phi$$. Consider a point $$\left( x_{0},y_{0}\right)$$ on $$F$$. Due to the strict monotonicity of $$\phi$$, the triplet $$\left( \left( x_{0},\phi \left( x_{0}\right) \right) ,\left( x_{0},y_{0}\right) ,\left( x_{0},y_{0}\right) ,\left( \phi ^{-1}\left( y_{0}\right) ,y_{0}\right) \right)$$ determines a unique rectangle through the point $$\left( x_{0},y_{0}\right) ,$$ with each side parallel to one of the axes and having at least two vertices on $$\phi$$. The fourth point $$\left( \phi ^{-1}\left( y_{0}\right) ,\phi \left( x_{0}\right) \right)$$ of the rectangle belongs to the set $F^{\phi }:=\{(x,y)\in \mathbb{R}^{2}| \left( \phi ^{-1}\left( y\right) ,\phi \left( x\right) \right) \in F\}.$ Call $$F^{\phi }$$ the $$\phi$$-inverse of $$F$$. The $$\phi$$-inverse of a function $$f$$ is again a function if and only if $$f$$ is injective. To other monotone functions, the authors associate a set of $$\phi$$-inverse functions.

MSC:

 26A48 Monotonic functions, generalizations

Keywords:

Real function; inverse; symmetry
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