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517 HW4 1. Let f : X → X, with X complete and f Lipschitz continuous with constant K < 1. Prove that there is x∗ ∈ X such that f (x∗ ) = x∗ , and that for each x ∈ X, {f n (x)} converges to x∗ . 2. Let f : R → R. We say f is uniformly differentiable if for each > 0, there exists δ > 0 such that 0 < |x−y| < δ implies f (y) − f (x) 0 < . − f (x) y−x Prove that f is uniformly differentiable if and only if f 0 is uniformly continuous. 3. Assume f : R → R is continuous and satisfies f (x + y) = f (x) + f (y) for all x, y ∈ R. (i) Prove that if f is differentiable, then f 0 is constant. (ii) Prove that f is differentiable by showing that f (x) = cx for some c ∈ R. 4. Let f 0 be differentiable on [a − h, a + h] such that f 00 is continuous at a. If f 0 (a) = 0 and f 00 (a) < 0, use Taylor’s theorem to show that f has a strict local maximum at a, that is, f (x) < f (a) for x in a neighborhood of a. If the assumption that f 00 is continuous at a necessary? Justify your answer with proof. 1