The Fixed Point Theorem

Theorem:

Let f be a continuous function on [0,1] so that f(x) is in [0,1] for all x in [0,1]. Then there exists a point p in [0,1] such that f(p) = p, and p is called a fixed point for f.

Proof:

If f(0) = 0 or f(1) = 1 we are done . So assume the points 0 and 1 are not fixed points. Let g(x) = f(x) -x. Then since 0 and 1 are not fixed points g(0) = a > 0 and g(1) = b < 0 . So by the Intermediate Value Theorem, there exists a point p so that g(p) = 0 . The point p is a fixed point for f .

Introductory Page