Definition:Minimum Value of Real Function/Local

Definition
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 minimum at $\xi$ iff:


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

That is, iff there is some subinterval on which $f$ attains a minimum within that interval.

Also see

 * Definition:Local Maximum
 * Derivative at Maximum or Minimum