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Let $\phi$ a differentiable function on $\Omega \in \mathbb{R}^n$, open connected set.
then if $\frac{\partial\phi}{\partial x_i}=0 \; for \; i=1,..,n$.
we have $\phi = constant$, because $\Omega$ is connected!
I didn't get why $\Omega $ should be connected.
If $U$ and $V$ are disjoint open sets, $\phi(x)=1$ for all $x \in U$ and $\phi(x)=0$ for all $x \in V$ then all the partial derivatives are $0$ but $\phi$ is not a constant function on $U \cup V$.
Since $\Omega$ is open and connected, any two points $x,y \in \Omega$ can be joined by a path with a finite number of segments that are parallel to the axes. Then the mean value theorem shows that $\phi(x)=\phi(y)$ and hence $\phi$ is constant on $\Omega$.
