Definition:Minimum Value of Real Function

Definition

Absolute Minimum

Let $f: \R \to \R$ be a real function.

Let $f$ be bounded below by an infimum $B$.

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 minimum value of $f$ on $S$, and this minimum is attained at $x$.

Local Minimum

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 $\xi$ if and only if:

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

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