Let $f\colon\mathbb R^n\to\mathbb R$ be a function, such that for all $x\in\mathbb R^n$ the partial derivative $(D_jf)(x)$ exists. Show the following: if $(D_jf)(x)=0$ for each $x$ and $j$, then $f(x)$ is constant.

So I began as follows: Assume $f$ is not constant. That means there are distinct $x_1,x_2\in\mathbb R^n$, such that $f(x_1)\neq f(x_2)$. Now I would like to use the mean value theorem along with the following function: consider $a\in\mathbb R^n$, and let $g_i(x)=f(a_1,\dots,a_{i-1},x,a_{i+1},\dots,a_n)$. We know that $(D_if)(a)$ exists iff $g_i'(a_1)$ exists. However, I'm kinda …