Approximate $f$ in $L^1$ by continuous $g$. The limsup local oscillation of $f$ is bounded by the maximal <function> of $f-g$ plus the vanishing oscillation of $g$. The weak $(1,1)$ <Hardy-Littlewood maximal inequality> makes the exceptional set have measure at most $C\|f-g\|_1/\epsilon$; let the approximation error tend to zero.

Solution

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

 Solution (source code)