Monotone Path: For any two points $x,y$, there exists an increasing path $\gamma$ with terminals $x,y$ and $f\circ\gamma$ is monotonic along the path.

Question

Consider a function $f: \Bbb R^n \to \Bbb R$ and points $x, y \in \Bbb R^n$. We say that $\gamma$ is an “increasing path from $x$ to $y$” if

• $\gamma: [0, 1] \to \Bbb R^n$ is continuous with $\gamma(0) = x$ and $\gamma(1) = y$, and
• $f \circ \gamma$ is increasing on $[0, 1]$.

Question: What assumptions must be made on $f$ to ensure that for every two points $x,y$ in $ \Bbb R^n$ there is an increasing path joining them?

Motivation: this problem is related to the gradient descent method in optimization, in which you want your $f\circ\gamma$ be increasing in every step. If such path exists, then optimization method works. If not, then the method does not work.

Ideas: for $n=1$, the things are trivial, and we need $f$ to be monotonic (not necessarily continuous)

For $n\geq2$, however, things become more interesting. It seems like convexity or concavity is sufficient. (Update: convexity is not sufficient, the new hypothesis now is strict convexity)

Note: there is also similar idea in graph theory called "monotone path"

Answer 1

So

1. Every two points needs to have path, so we need absolute continuity.$f(x,y)= tg x $ is continuous in own domain,but for example there's no path between $(0,0,0)$ and $(\pi, 0, 0).
2. More accurate clue than is existence of partial extreme points than convexity itself