Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 22H / c

At a Lebesgue point, $f\ast g_n(x)-f(x)=\int g(u)[f(x-u/n)-f(x)]du$. Compact support and boundedness of $g$ reduce this to a constant times the mean oscillation over a ball of radius $O(1/n)$, which tends to zero.