A function f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). In other words it says that if f is continuous on [a,b] and if y0 is a value between f(a) and f(b), then y0=f(c) for some c that is between a and b. This theorem is the reason the graphs of a continuous function on an interval can not have any breaks or holes.
The Intermediate Value Theorem tells us that...
The Intermediate Value Theorem actually tells us a story of despair. A story about a boy who foreseeably fails the final part of numerous unsuccessful attempts to study mathematics. In the end, he cuts his wrists with an integral symbol shaped blade. But instead of dying, he just becomes an emo. How appropriate.