MTH 433 University of Miami Advanced Calculus Questions

5 questions you need to solve

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1. [10 points] Mark each statement true or false (you do not need to justify your answer).(i) Let f : D ? R and let c ? D, if c is an isolated point of D, then f is continuous at c.(ii) Let f : [a, b] ? R be continuous and suppose f(a) ? 0 ? f(b). Then there exists a point c in(a, b) such that f(c) = 0.(iii) Let f : D ? R be a continuous function, if D is bounded, then f(D) is bounded.(iv) In the definition of uniform continuity, the positive ? depends only on the function f andthe given ?.(v) Let f : D ? R be a uniformly continuous function, if D is bounded, then f(D) isbounded.2. [10 points] Let f : D ? R be continuous at c ? D and suppose that f(c) > 0. Prove that thereexists a neighborhood U of c such that f(x) > 0 for all x ? U ? D.3. [10 points] Find a ? > 0 so that |x?1| < ? implies that |?x2 + 1??2| <110 . Justify your answer.4. [10 points] Prove that the function f(x) = 1xis uniformly continuous on [2, ?) by verifying the?-? property in Definition 5.4.1 in textbook (i.e. Definition 4.1 in Section 5.4 Notes)