extreme value theorem

On a finite, closed stretch of road, a continuous landscape has an actual highest point and an actual lowest point — not just heights it gets near, but heights it truly reaches. The extreme value theorem guarantees that a continuous function on a closed bounded interval attains a genuine maximum and a genuine minimum.

Precisely: if f is continuous on a closed and bounded interval [a, b], then there exist points c and d in [a, b] with f(c) <= f(x) <= f(d) for all x in [a, b]. The function is first bounded (the boundedness theorem), and then the supremum and infimum of its values are not merely approached but actually attained.

Every hypothesis is load-bearing. Drop closedness: f(x) = x on the open interval (0, 1) attains neither a max nor a min. Drop boundedness of the interval: f(x) = x on [0, infinity) has no maximum. Drop continuity: a function with a jump on [0, 1] can fail to reach its supremum. The clean statement in topology is that a continuous real function attains its extrema on any compact set.