Definition:Minimum Value of Real Function/Local

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.