Hill climbing extremist

Solution 1:

I haven't thought this through, but at the same time it would take too much space as comment, so let me describe an idea I have for this problem.

$\newcommand{\R}{\mathbb{R}}\newcommand{\vp}{\varphi}$ Idea. The relation $x \sim y$ if there is a path joining $x$ and $y$ is an equivalence relation in $\R^2$. If you can only prove that each equivalence class is open, then your claim follows from connectedness of $\R^2$.

Other observations:

• I would expect that one needs some assumptions on behavior of $\vp$ in infinity, so it could be easier to work on a torus or a sphere first, and then see what is needed in the case of $\R^2$.
• It is surely important that our vector field is a gradient field. On a torus any nonzero constant vector field yields a counterexample, so one probably needs to use the growth of $\vp$ in some way.
• To make things clearer, I would add an additional assumption that $\vp$ is a Morse function. This way we can be sure what happens at points where $\nabla \vp = 0$.