Question

A fixed point of a function f is a number c in its domain such that . (The function doesn't move c; it stays fixed.)

(a) Sketch the graph of a continuous function with domain whose range also lies in . Locate a fixed point of f.

(b) Try to draw the graph of a continuous function with domain  and range in that does not have a fixed point. What is the obstacle?

(c) Use the Intermediate Value Theorem to prove that any continuous function with domain and range in must have a fixed point.