In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This has two important corollaries: 1) If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). 2) The image of a continuous function over an interval is itself an interval. == Motivation == This captures an intuitive property of continuous functions...