Definition:Local Minimum

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Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then $f$ has a local minimum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$

That is, if and only if there is some subinterval on which $f$ attains a minimum within that interval.


Note the requirement for the intervals to be open.

A closed interval of course includes the value of $f$ at its end points and so every closed interval attains a minimum.

Also see