If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$.
How can we prove that $f$ does not depend on $x_1$? It is not clear to me why the domain has to be convex and what if $U$ is just path wise connected? I would appreciate some help. Thank you!
Start with the simple case of $U \subset \mathbb{R}$. If $U$ is a single interval and $f: U \to \mathbb{R}$ satisfies $f'(x) = 0$, do you see why $f$ must be constant? If $U$ has multiple connected components, do you see why $f$ does not have to be a constant function?
Now, in the case $U \subset \mathbb{R}^n$, apply the above results to $f$ restricted to the line starting at $(a,x_2,x_3,...,x_n)$ and ending at $(b,x_2,x_3,...,x_n)$. Where is convexity being used?