Definition:Minimum Value of Real Function/Local

Definition
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 $\xi$ :


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

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

Warning
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 minimum.

Also see

 * Definition:Local Maximum
 * Derivative at Maximum or Minimum