Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$
Then how to prove that $f$ is a constant? I had this exercise in a class last year but I couldn't find how to do the induction step (for $n = 1$ it is clear of course).