Calculus on Cantor triadic set. I: Derivative (Q1377189)

The author introduces the derivative of a real function defined on the classical Cantor triadic set \(\mathbf C\). Namely, denoting by \(\varphi\) the Cantor function associated to \(\mathbf C\), the function \(f:{\mathbf C}\to{\mathbf R}\) has a derivative at the point \(x_0\in{\mathbf C}\) if there exists the limit \[ \lim_{x\to x_0, x\in {\mathbf C}}{{ f(x)-f(x_0) }\over { \varphi(x)-\varphi(x_0)}} \] With respect to this derivative a Newton-Leibniz-type theorem is proved. As a concrete example, it is studied the derivative of a self-similar function defined on the Cantor set \(\mathbf C\). Moreover, the subset of \(\mathbf C\) containing points at which the derivative does not exist is determined, and called exceptional set.