Rectifiability of the jump set of locally integrable functions

In this note we show that for every measurable function on ${\mathbb{R}}^n$ the set of points where the blowup exists and is not constant is $(n-1)$-rectifiable. In particular, for every $u\in L_{loc}^1({\mathbb{R}}^n)$ the jump set $J_u$ is $(n-1)$-rectifiable.