I have a problem where it seems like I should be able to visualize an answer, but I can't. Perhaps I need to take a more formal approach.
"Let $A$ be a non-empty open convex subset of $\mathbb{R}^2$, and suppose that $f: A \rightarrow \mathbb{R}$ satisfies that $\frac{\partial f}{\partial x} = 0$ at all points in $A$. Prove that there is a function $g$ of one variable such that $f(x,y) = g(y)$. Show that this conclusion may fail if convexity is replaced by connectedness."
For the first part, maybe I can do something like the following:
When $A$ is convex, I can take any points $\boldsymbol{a_0}, \boldsymbol{a_1} \in A$, where $\boldsymbol{a_0} = (x_0, y_0), \boldsymbol{a_1} = (x_0 + h, y_0)$, and have the line segment connecting them be in $A$.
Since $\frac{\partial f}{\partial x} = 0$ everywhere on this line segment, some mean value business shows that $f(\boldsymbol{a_0}) = f(\boldsymbol{a_1})$, regardless of the value of $h$. Therefore, at any given $y=y_0$, $f$ does not depend on $x$, so $f(x,y) = g(y)$ only.
Obviously I used convexity of $A$ to show this, but I can't seem to find a counterexample that makes it fail when $A$ is connected but not convex. If $A$ were not even not connected, so it was separable into disjoint open sets $A_1$ and $A_2$, it would be easy to see that $f$ could have different constant values w.r.t. $x$ in each separate region. This separation could allow $f$ to depend on $x$ overall even though $\frac{\partial f}{\partial x} = 0$ where ever $f$ is defined.
It's the connected but not convex case that's troublesome. The more I think about it, the less sensible it sounds. If we want to show that $f(x,y)$ is not just a function of $y$, we should be able to find some $\boldsymbol{a_0}, \boldsymbol{a_1} \in A$, as above (having the same $y$ values but different $x$ values) such that $f(\boldsymbol{a_0}) \neq f(\boldsymbol{a_1})$. We suppose $A$ is connected. Since $\frac{\partial f}{\partial x} = 0$ everywhere in $A$, it seems like any curve in $f(A)$ that connects those points has to lie in a plane parallel to the $xy$ plane, else $f$ would change with changing $x$. But if that were so, how could we have $f(\boldsymbol{a_0}) \neq f(\boldsymbol{a_1})$?
Clearly my intuition is failing me here. Could somebody set me straight?
This is my first post on Math Exchange, please let me know if I'm doing something wrong. Thanks!
