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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!

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    $\begingroup$ You won't get anything nice just by demanding a single sublevel set is path-connected--this condition is very weak. Maybe something interesting can be said if you demand that ALL of the sublevel sets are path-connected. $\endgroup$
    – Jack Huizenga
    Commented Oct 10, 2012 at 0:26

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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. 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.

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