= Differentiability almost everywhere of supercritical Sobolev functions
For $p>n$, the <Hölder continuous function> representative $u^*$ of $u\in W^{1,p}(\mathbb R^n)$ has a classical <Fréchet derivative> at almost every point, equal to its <weak derivative>. At a point $x$ where the $p$-mean oscillation of $Du$ tends to zero, apply the <Morrey inequality on a cube> to $u^*(y)-u^*(x)-Du(x)\cdot(y-x)$. On a cube of side comparable to $|y-x|$, its oscillation is at most
$$
C|y-x|\left(\frac1{|Q_r(x)|}\int_{Q_r(x)}|Du(z)-Du(x)|^p dz\right)^{1/p}=o(|y-x|).
$$
The required points have full measure by the <Lebesgue differentiation theorem>.
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