Suppose that a function f is continuous on the closed interval [0, 1] and that 0...

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Suppose that a function f is continuous on the closed interval [0, 1] and that 0 ≤ f(x) ≤ 1 for every z in [0, 1]. Show that there must exist a number c in [0, 1] such that f(c) = c (c is called a fixed point of f). Suppose that a function f is continuous on the closed interval [0, 1] and that 0 ≤ f(x) ≤ 1 for every z in [0, 1]. Show that there must exist a number c in [0, 1] such that f(c) = c (c is called a fixed point of f).