Limit of a continuous Sobolev function to its boundary

Let $U$ be an open bounded subset of $\mathbb R^n$ and $\partial U$ be Lipschitz.
By this answer, for $u \in W^{1, p}(U)$,$$\lim_{r \rightarrow 0} \frac{1}{|B_r(x_0)\cap U|} \int_{B_r(x_0)\cap U} |u(y) - u(x_0)|^p = 0$$for $\mathcal{H}^{n-1}$-a.e. $x_0 \in \partial U$.

Now, suppose that $u \in W^{1, p}(U) \cap C(U)$. Then does the following stronger statement holds?:$$\lim_{x \rightarrow x_0, x\in U} u(x) = u(x_0)$$for $\mathcal{H}^{n-1}$-a.e. $x_0 \in \partial U$.