 # 6. (10 points) A function f D R is said to be uniformly approxi...

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6. (10 points) A function f D R is said to be uniformly approximated by polynomials if for every e > 0 we can find a polynomial p(x) such that If(x) p(x) < € for all x € D. Show that the following two functions cannot be uniformly approximated by polynomials: (a) The function f (0,1) -> R given by f(x) = : (b) The function g R -> R given by g(x) = 1+x² Hint: For part (b) you may use that a polynomial on R is bounded if and only if it is constant. 7. (15 points) (a) Define what it means for a sequence {fn D -> R} to converge uni- formly to the function f D -> R. (b) For each n € N, let fn R R be given by fn(x) = Vx2+# Prove that the sequence {fn} converges uniformly on R to the function f R -> R given by f(x) = |x|. (c) Give an example, with proof, of a domain D and a sequence of func- tions which converges pointwise, but not uniformly on D. 8. (10 points) For each n € N let fn R - R be given by fn(x) = En-on Let f R + R be given by f(x) = e2. Let r be any positive real number. Prove that {fn} converges uniformly to f on [-r,r]. Hint: Use the Lagrange Remainder Theorem. You may also use that limn-+0 rth n! = 0 (Lemma 8.13).

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