Definition:Minimum Value of Real Function/Local/Strict

Definition

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

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

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

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

A strict local minimum is also known as a strict relative minimum.

Some sources refer to this as a local minimum and do not consider the situation where $\forall x \in \openint c d: \map f \xi \le \map f x$.

Some sources assume local as given, and merely refer to this as a minimum or strict minimum.