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Let f : E → R be a differentiable function where E = [a,b] or E...

Let f : E → R be a differentiable function where E = [a,b] or E = (−∞,∞), show that if f′(x) not = 0 for all x ∈ E then f is one-to-one, i.e., there does not exist distinct points x1,x2 ∈ E such that f(x1) = f(x2). Deduce that f(x) = 0 for at most one x.

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