Maximum and minimum

Remarks

The maximum value of $f$ is the greatest element of the image $f(D)$, when it exists. Then it equals the supremum of $f(D)$, and the supremum is attained. The maximum value is unique, but it can be attained at several points. For $D \subseteq \R^n$, every global extremum is a local extremum.

Example: $f(x) = x^2$ on $D = [-1, 2]$ attains its maximum at $a = 2$. Indeed, for $x \in [-1, 2]$ one has $\abs{x} \le 2$, so $x^2 \le 4 = f(2)$.

Non-example: $f(x) = x$ on $D = (0, 1)$ has no maximum. For each $a \in (0, 1)$, the point $x = (a + 1)/2$ lies in $D$ and $f(x) > f(a)$. The supremum of $f(D)$ is $1$, but it is not a value of $f$. The extreme value theorem gives a maximum and a minimum for every continuous function on a closed bounded interval.