MATH 410 Study Guide - Final Guide: Weierstrass Factorization Theorem, Uniform Convergence

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10 Jan 2019
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The nal exam will be worth 200 points, of which this takehome part will be 45 points. This takehome part is due when you come to take the in-class nal: (10 points) for x > 0 and x 6= 1, de ne f (x) = x1/(x 1). Justify your answer: (5 points) we"ve proved that if fn is a sequence of integrable functions on [0,1] converging uniformly to a function f , then. Prove that the sequence fn converges uniformly on [0,1]: (15 points) the weierstrass theorem showed that any continuous function on a closed bounded interval is a uniform limit of polynomials. In contrast, prove the following: if f : r r and f is a uniform limit of polynomials, then f is a polynomial.

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