My favorite theorem is Rolle's Theorem from calculus. It states, if a real valued function is continous on a closed interval [a,b], differentiable on the open inverval (a,b), and f(a)=f(b), then there is some real number c in the open interval (a,b) such that f'(c)= 0. In other words: a differentiable function, which attains equal values at two points, must have a point somewhere between them where the slope is zero.
