 Solution

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 $C\Vert f-g{\Vert }_1/ϵ$; let the approximation error tend to zero.