(a) Let ƒ : [a, b] → R be continuous with f(a) < 0 and f(b) > 0, and S := {x = [a, b] | f(x) < 0}. If c is the supremum of S, prove that f(c) = 0. (b) Suppose that ƒ : (0,1) → R is uniformly continuous. Prove that limx→o+ f(x) exists.

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