Definition 4

We say that a real function $f$, defined on a subset A of a topological space E, has a relative maximum (resp. strict relative maximum, relative minimum, strict relative minimum) at a point $x_0 \in A$, relative to A, if there is a neighbourhood $V$ of $x_0$ in E such that at every point $x \in V \cap A$ distinct from $x_0$ one has $f(x) \leq f(x_0)$ (resp. $f(x) < f(x_0)$, $f(x) \geq f(x_0)$, $f(x) > f(x_0)$).

It is clear that if $f$ attains its least upper bound (resp. greatest lower bound) over A at a point of A, then it has a relative maximum (resp. relative minimum) relative to A at this point; the converse is of course incorrect.

Note that if $B \subset A$, and if $f$ admits (for example) a relative maximum at a point $x_0 \in B$ relative to B, then $f$ does not necessarily have a relative maximum relative to A at this point.