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!



When is a sublevel set path-connected?
If $f$ is $C^1$ and coercive, the existence of a non-path connected sub-level set implies the existence of a critical point, e.g. by the [mountain pass theorem](http://en.wikipedia.org/wiki/Mountain_pass_theorem). If $f$ is $C^2$, the Hessian of this critical point has a "strong" Morse index $m \_ \* $ (number of negative eigenvalues) and a "weak" Morse index $m ^ \* $ (number of non-positive eigenvalues) related by $m \_ \* \le1\le m ^ \*$. Therefore, on the lines of Jack Huizenga's comment above, if $f$ is known to have no such critical point, beside being coercive and smooth, we may conclude that all sub-level sets are path connected. 


