On a canonical parametrization of continuous functions. (English) Zbl 0779.39002

Neuman, František

Zesz. Nauk. Akad. Górn.-Hutn. Stanisł. Staszica 1335, Opusc. Math. 6, 185-191 (1990).

The author introduces the following definition. Two continuous functions \(f\) and \(g\) are said to be \(k\)-equivalent \((k\in\mathbb{N})\), if there exists a diffeomorphism \(\varphi\) of \(\text{Dom}(g)\) onto \(\text{Dom}(f)\) such that \(\varphi'\neq 0\) on \(\text{Dom}(g)\) and \(f\circ\varphi=g\). Then he proceeds to study the following problem. Select from each class of equivalence just one representative function in such a manner that the following hereditary condition is satisfied: if \(h:I_ 0\to J_ 0\) is a representative of the class to which a function \(f:I\to J\) belongs and \(\varphi\) is a \(C^ k\)-diffeomorphism for which \(f\circ\varphi=g\) holds, then \(h\) restricted to \(i_ 0\subset I_ 0\) is the representative of the class to which the restriction of \(f\) to \(\varphi(i_ 0)\) belongs.

Reviewer: J.Matkowski (Bielsko-Biała)

MSC:

39B22

Functional equations for real functions

Keywords:

continuous functions; diffeomorphisms; \(k\)-equivalence classes