If $f:[a,b]\to\Bbb R$ is continuous and non-constant, prove that at least one $x_0\in [a,b]$ is NOT a local extremum.
This proposition isn't as trivial as it seems. For example, in the case of Cantor function, $f$ has zero derivative a.e., therefore almost every $x\in[0,1]$ is a local extremum for $f$. In fact, I'm not even able to show it doesn't stand as a counterexample to the proposition. (By monotonicity, each rational point in the Cantor set is also a local extremum for the Cantor function, since it borders on an open interval where the function has zero derivative. So the only candidates for non local extrema are among the "bad" (irrational) points, which are hard to analyse.)
Thanks for your help.