\begin{definition}[Definition:Minimum Value of Real Function/Local]
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 $\xi$''' {{iff}}:
:$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$
That is, {{iff}} there is some subinterval on which $f$ attains a minimum within that interval.
\end{definition}