No-where averagely differentiable functions: Baire category and the Takagi function

Abstract

A classical and well-known result due to Banach and Mazurkiewicz says that a typical (in the sense of Baire) continuous function on the unit interval is no-where differentiable. In this paper we prove that a typical (in the sense of Baire) continuous function \(f\) on [0, 1] is spectacularly more irregular than suggested by Banach and Mazurkiewicz’s result. Namely, not only is the difference quotient \(\frac{f(x+h)-f(x)}{h}\) divergent as \(h\rightarrow 0\) for all \(x\), but the function \(h\rightarrow \frac{f(x+h)-f(x)}{h}\) diverges so badly as \(h\rightarrow 0\), that it remains divergent even after being “smoothened out” using iterated Cesaro averages of arbitrary high order. More precisely, we introduce the notion of higher order average differentiability based on iterated Cesaro averages and prove that not only is a typical (in the sense of Baire) continuous function on [0, 1] no-where differentiable, but it is even no-where averagely differentiable of any order. We also show that the no-where differentiable Takagi function is, in fact, no-where averagely differentiable.