
Analysis in Rn
I have tried to solve this question and I dont find a solution. Can you please help me? Thanks. ((Thinking) I dont know how to display the problem in latex)
\noindent a) Suppose $f: U \to \rr$ where $U$ is an open convex subset of $\rr^n$ and that $(\partial f/ \partial x_1)(p) = 0$ for all $p \in U$. Prove that $f$ depends only on $x_2, x_3, \ldots, x_n$ \\
\noindent b) Consider the region $U = \rr^2 \setminus L$ where $L$ is the halfline $\{(x, y): x = 0 \text{ and } y \ge 0\}$. Find a funciton $f: U \to \rr$ such that $(\partial f / \partial x)(p) = 0$ for all $p \in U$ but it is not true that $f$ depends only on $y$.