If $f \in C^k(\Omega)$, with $\Omega\subset \mathbb{R}^n$ open and such that $\partial \Omega$ is regular, than by Gauss-Green formulas, we can show that the classical $\alpha$ derivative $D^\alpha f$ of $f$ coincides with the distributional one, for all $\alpha$ multi index s.t. $|\alpha| \le k$. Does it exist a function $f:\mathbb{R}^n \to \mathbb{R}$, with $n>1$, differentiable but not continuously differentiable s.t $D^\alpha$ is integrable but it doesn’t coincide with the distributional one?
In other words, does it exist a Sobolev function (defined on $\mathbb{R}^n$ with $n>1$) whose distributional derivative do not coincide with the classical one?