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Elementary Analysis 1. Consider a set X equipped with the distance d defined by d(x.y) = 0 if x=y and a metric space Y. Determine all continuous maps from x to Y. 2. Let (X,d) be a metric space and a € X. Assume that u, U and f are real-valued functions on X such that for all . € x and that lim u(x) = e S " = lim v(x). Prove that for any E > 0, there exists r >0 such that f(x) E(l-s,CtE) for any x € B(a,r). 3. The indicator function of Q is defined on R by = Determine all the points where Xg is continuous. 4. Let f IR - R be a function that is continuous at 0 and additive, that is, f (I for every x. y, € IR, (a) Prove that f(nx) = /(x) for any n €Z and any x € R. (b) Prove that f(r) = f(1) for any r € Q. (c) Prove that f is continuous on R and derive an expression of f(x) for any I € R. 5. The function E is defined on IR by E(x) = where is the only integer n € Z satisfying 72 SoThe purpose of this problem is to study the function defined on IR by if 0 (a) Calculate f(x) for r < - 1 and for x > 1. Let N*. (b) Determine f(x) for I € (P+ and verify that f is continuous on every such interval. 1 (c) Study the continuity of f on the right of p (d) Is f continuous at 0? Hint: prove that -

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