I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (e.g. $f$ concave), but is there any suite of conditions that is necessary and sufficient?

We can assume $f$ is continuous.

If you have any ideas or recommended reading, I'd love to hear about it. Thanks!