Boundedness Theorem - Expii

The boundedness theorem says that if a function f(x) is continuous on a closed interval [a,b], then it is bounded on that interval: namely, there exists a constant N such that f(x) has size (absolute value) at most N for all x in [a,b]. This is not necessarily true if f is only continuous on an open (or half-open) interval: for instance, 1/x is continuous on the open interval (0,2018], but it is unbounded. Anyways, the boundedness theorem is a special case of the more important extreme value theorem, which we'll discuss next.