+99 Prove: If f is continuous on the interval [0, 1] and 0 ≤ f(x) ≤ 1 for all x, then there is a number c with 0 ≤ c ≤ 1 such that f(c) = c. (The number c is called a “fixed point” of f because the image of c is the same as c: f does not “move” c.) Hint: Define a new function g(x) = f(x) − x and start by considering the values g(0) and g(1).
