Definition:Maximum Value of Real Function
Let $f: \R \to \R$ be a real function.
It may or may not be the case that $\exists x \in \R: \map f x = B$.
If such a value exists, it is called the maximum of $f$ on $S$, and that this maximum is attained at $x$.
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.