A fixed point of a function f is a number in its domain such that f(c) = c. (The function doesn’t move ; it stays fixed.)
(a) Sketch the graph of a continuous function with domain [0, 1] whose range also lies in [0, 1]. Locate a fixed point of f.
(b) Try to draw the graph of a continuous function with domain [0, 1] and range in [0, 1] 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 [0, 1] and range a subset of [0, 1] must have a fixed point.