On the problem of characterizing derivatives
The author deals with the question of a characterization of derivatives. Considerations of a philosophical character and historical comments are used to reach a conclusion that a solution of the problem should contain an integral. To support this view the author presents the following: {Theorem}. Let \(f\) be a derivative with \(f(0)=f(1)=0\). Then \(\int _0^1 f(x)dx=L\) if and only if \(g_L(x)\) is a derivative. Here \(g_L(x)=0\) for \(x\leq 0\), \(g_L(x)=L\) for \(x\geq 1\) and \(g_L(x)=f(h(x))\) for \(0<x<1\), where \(a_n=1-1/n\) and for each positive integer \(n\) \[ h(x)=\frac {x-a_n}{a_{n+1}-a_n}\text{ if } x\in [a_n,a_{n+1}]. \] Then a slight modification of the Kurzweil-Henstock integral is used to characterize derivatives: {Theorem}. A function \(f\) is a derivative if and only if for all \(\varepsilon >0\) there exists a \(\delta :\mathbb{R}\rightarrow \mathbb{R}^+\) such that for all intervals \(I\subset \mathbb{R}\) any two \(\delta \)-fine tagged partitions of \(I\) have Riemann sums which differ by less then \(\varepsilon |I|\). The author argues that an object or procedure needed for characterization of derivatives should be at least as complicated as an integral.
scientific article; zbMATH DE number 1449466