Definition:Local Maximum

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Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi \in \left({a \,.\,.\, b}\right)$.

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

$\exists \left({c \,.\,.\, d}\right) \subseteq \left({a \,.\,.\, b}\right): \forall x \in \left({c \,.\,.\, d}\right): f \left({x}\right) \le f \left({\xi}\right)$.

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

Also see