The question is simple but I still can't prove it or contradict it. Here it goes:

Suppose $f$ and $g$ are defined on the circle (or, equivalently, $2\pi$ periodic functions) and Lebesgue integrable, is their convolution $(f*g)(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x-y) g(y) dy $ continuous?

In the case when two functions are bounded, it is proved in Elias Stein's
*Fourier Analysis* (page 47) that their convolution **is** truly continuous.
However, for unbounded functions, I have tried tools in real analysis, say, Lusin's theorem, transition continuity of $L_1$ functions, etc., but can't figure it out.