Question on limit in $L^p$

Question

Let $\{f_n\}$ be differentiable real-valued on $[0,1]^2$.

Assume $\nabla f_n$ is convergent in $L^p$.

Then, is there $f$ s.t. $\nabla f=\lim_{n\rightarrow\infty}\nabla f_n$ almost everywhere?

(You may choose any $p>1$ as you want)

Answer 1

There exists a sequence of continuous funtions $\{g_n\}$ on $[0,1]$ converging in $L^{p}$ to a function $g$ but not converging almost everywhere. [ MSE has such examples]. Let $f_n(x,y)=(\int_0^{x}g_n(t)\, dt,0)$. This gives a counter-example since $\int_0^{x}g_n(t)\, dt \to \int_0^{x}g(t)\, dt$ in $L^{p}$ (an easy consequence of Holder's inequality).