I'm trying to disprove/prove the following claim:

Let $f:[0,1] \to R$ be a continious function, such that:

$$f(0.5) < f(0) < f(1) $$

Prove that there exists a local minima or maxima to $f$.

It seems so simple to prove. Since its continuous then $f(0)$ is a maxima or otherwise there must be a point near it which is, since its point which is lower then others in the given region.

Unfortunately, I failed to figure out a formal proof.

  • $\begingroup$ A much more general result holds. Having a continuous function on a compact domain alone grants the existence of local minima and maxima without any additional conditions. The name of this result is "extreme value theorem". $\endgroup$ – joedoe8585 Jan 23 '16 at 17:46

Since $f$ is continuous on $[0,1]$, it has at least one global minimum in this interval. Since $f(0.5)\lt f(0)$ and $f(0.5)\lt f(1)$, any such global minimum is not at an endpoint, so it must be a local minimum.

  • $\begingroup$ I can understand it by reason , but i don't feel its formal proof yet. $\endgroup$ – John Schwartz Jan 23 '16 at 19:04
  • $\begingroup$ We have used but not proved the fact that a continuous function on a closed bounded interval has a global min (and global max). This is a very standard theorem that you probably already have. The rest is basically fully justified in the above answer. For it is clear from the definition of local min that a global min attained in the interior of our interval is a local min. $\endgroup$ – André Nicolas Jan 23 '16 at 19:08

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