I want to ask about example of real valued functions defined on the real line such that their convolution exist in every point and is discontinuous on a "large" set, for example on each point of some interval or in dense subset of $\mathbb{R}$ or maybe on the whole $\mathbb{R}$.
My question is related to paper Mikusinski, Ryll-Nardzewski, Sur le produit de composition, http://matwbn.icm.edu.pl/ksiazki/sm/sm12/sm1213.pdf
The authors consider convolution of integrable functions which are zero for $x\leq 0$. On page 52 the above paper is given example of two integrable functions such that their convolution is discontinuous at a point. Autors say also, but there is no proof of this statement, that it is possible by condensation of singularities to construct integrable functions such that their product is discontinuous everywhere (on $\mathbb{R_+}$).
Thanks.
