Definition:Local Maximum

Definition

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 maximum at $\xi$ if and only if:

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

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

Warning

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 maximum.