I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P)

*For any $x, y \in U$ such that $f(x)<f(y)$, there exists a continuously differentiable path $\gamma:[0,1] \rightarrow U$ such that $ \gamma(0)=x, \gamma(1)=y$ and $f(\gamma(t))$ is increasing.*

Intuitively, this "there is always a way up" property will hold for $f$ such that $f^{-1}((c, \infty))$ is connected for any $c \in \mathbb R$. However, if the set $f^{-1}(c)$ is highly irregular then my intuition might break down.

So the question is, does this criteria guarantee the property (P)? If not, is there a simple criterion for $f$ such that (P) is satisfied?

**Added:** Since the question is for satisfying my curiosity rather than solving a specific problem, feel free to modify any hypothesis of the question (changing 'connected' to 'path connected' or changing the regularity of $f$ for example). Any result, positive or negative, is welcomed.

**Added:** The criterion I proposed has a trivial counterexample: consider $f$ on $\mathbb R$ having a unique maximum. Anyway, the main question is to find a criterion for (P). I guess it should be that the inverse image of any interval is path connected.

pathconnected, if we want to make such an assumption? $\endgroup$1more comment