Let $f\in W^{1,2}_{\text{loc}}(\mathbb R^2)$. Here, $W^{1,2}_{\text{loc}}(\mathbb R^2)$ denotes the usual Sobolev space. More explicitly, $f:\mathbb R^2\to\mathbb R$ is a function such that, for every relatively compact open set $U\subset\mathbb R^2$,

- $f\vert_U\in L^2(U)$ ;
- there exist $g_1,g_2\in L^2(U)$ such that $$\int_U f\partial_1\phi=-\int_U g_1 \phi,\text{ and }\int_U f\partial_2\phi=-\int_U g_2 \phi$$ for all test functions $\phi\in C_{\text{c}}^\infty(U)$.

**My question.** Is the function $F:\mathbb R^2\to\mathbb R$, defined by
$$F(x,y)=\int_0^y f(x,t)\,\mathrm dt$$
continuous?

More precisely stated, does there exist a function $\tilde F\in C(\mathbb R^2)$ such that $F=\tilde F$ Lebesgue-almost everywhere? (Note that the function $F$ is not well-defined at every point since $f$ is only defined as an equivalence class modulo "being equal almost everywhere".)

Note that (cf. Brezis *Functional Analysis, Sobolev Spaces and Partial Differential Equations*, Lemma 8.2) the function $F$ is continuous in the variable $y$ and, weakly, $\partial_2 F=f$. However, I don't even see whether $F$ needs to be continuous in $x$.

*Remark.* If we had for instance $f\in W^{1,3}_{\text{loc}}(\mathbb R^2)$, then it would be clear that $F$ is continuous, since, by Morrey's inequality (see Evans *Partial Differential Equations*, chapter 5.6.2, Theorem 4), the space $W^{1,p}(\mathbb R^n)$ can be embedded into $C^0(\mathbb R^n)$ whenever $p>n$. But my case is $p=n$, so this Theorem doesn't apply.