Let $f:ℝ^{m}→ℝ$ is a continuous function. Let us consider the set $$A=\{x∈ℝ^{m} \ |\ f(x)=0\} = f^{-1}[\{0\}]$$

My question is: Under what conditions the set $A$ is connected?, simply connected?

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    $\begingroup$ Convexity might be relevant here. Not sure though. $\endgroup$
    – shalop
    Sep 22, 2017 at 9:28
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    $\begingroup$ The exact conditions are of course whenever $f$ has the property that $f^{-1}(0)$ is connected/simply connected. I doubt there is some other meaningful property of $f$ that is equivalent to that... $\endgroup$
    – skyking
    Sep 22, 2017 at 9:31
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    $\begingroup$ Convexity of $f$ is not enough. For example $f(x)=x^2-2$ would be counterexample. In higher dimensions, this example does not work though. $\endgroup$ Sep 22, 2017 at 9:32
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    $\begingroup$ Not sure if we can conclude (any topological properties of $A$) if $m > 1$ and if $f$ isn't bijective, if I'm wrong I'm really interested to see a counter example $\endgroup$ Sep 22, 2017 at 9:45