Non-Differentiability of the Convolution of Differentiable Real Functions

Abstract

We provide an example of two \(2\)-periodic everywhere differentiable functions \(f,g\colon\mathbb{R}\to\mathbb{R}\) whose convolution \(f*g\) fails to be differentiable at every point of some perfect (thus, uncountable) set \(P\subset\mathbb{R}\). This shows that the convolution operator can actually destroy the differentiability of these maps, rather than introducing additional smoothness (as it is usually the case). New directions and open problems are also posed.