Function problem

1. Suppose that the function f(x) is continuous in the closed interval [0, 1], and that 0 <= f(x) <= 1 for every x 2 [0, 1]. Show that there exists x [0, 1] such that f(c) = c.

2.Prove that the equation ex = 2 − x has at least one real root.
Update: 3. show that f(x) is not differentiable at x=0 where f(x) = x/ |x|
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