boundedness theorem
If you walk a continuous path across a finite, fenced-in field, you cannot climb to infinite height — the terrain stays within some maximum and minimum elevation. The boundedness theorem makes this precise: a continuous function on a closed bounded interval can never run off to infinity; its values stay trapped between two finite walls.
Precisely: if f is continuous on a closed and bounded interval [a, b], then f is bounded — there is a number M with |f(x)| <= M for all x in [a, b]. This is the first half of the road to the extreme value theorem, which strengthens ‘bounded’ to ‘attains its bounds’.
Both hypotheses are essential. On the open interval (0, 1], the continuous function f(x) = 1/x is unbounded; closedness fails at 0. On the unbounded interval [0, infinity), f(x) = x is unbounded; the interval has no right wall. The deep reason it works is compactness: a continuous image of a compact set is compact, hence bounded — the interval being closed and bounded is exactly what makes it compact in the real line.
f(x) = 1/x on (0, 1] is continuous but unbounded: as x -> 0+, f(x) -> +infinity. The half-open interval is not closed, so the theorem does not apply.
Lose closedness and continuity no longer forces boundedness.