If $f$ is a real-valued function defined in a convex open set $E$ of $\mathbb{R}^n$, such that $\frac{\partial f}{\partial x_1}$ is $0$ for every $x\in E$, prove that $f(x)$ does not depend on the coordinate $x_1$. Why do we need convexity of $E$ in order to show this?
Real functions of n variables
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real-analysis