continuous weak derivative $\Rightarrow$ classic derivative ?

Hello! I'm new in this formus, so I'm sorry if I make some mistake..

Let $u$ be a __weakly__ differentiable function on $\Omega\subset\R^n$ open bounded.

If $grad u$ is continuous, can I say $u\in\C^1(\Omega)$, i.e. $u$ __strongly__ differentiable? (better: u a.e. equal to a $C^1$ function)

Reading Evans' book the answer seems to be yes, but it's not so obvious to me.

I made a proof in the case $n=1$, but I'm not able to generalize it. I sum it up:

Since $C^\infty(\bar\Omega)$ is dense in $W^{1,1}(\Omega)$, you can show that for a.e. $a\in\Omega$

$u(x)=u(a)+\int_a^x u'(t) dt$ for a.e. $x\in\B(a,r)\subset\Omega$.

But if $u'$ is continuous, the integral function is $C^1$. Thus $u$ is a.e. equal to a $C^1$ function on $B(a,r)$.

Conclude by arbitrariety of $a$.

The problem is that in dimension $n>1$ I don't manage anymore to write $u$ as a $C^1$ function. Can you help me?

Can you generalize the proof for $n>1$, or do you have another idea to prove this result?