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

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.

### MSC:

 39B22 Functional equations for real functions