# final - Intro to Modern Analysis Summer MMXI Final Exam...

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Intro to Modern Analysis Summer MMXI Final Exam Name DIES LVNAE AD VII KAL AVG MMDCCLXIV * Do all problems, in any order. Explain all answers. An unju ified response alone may not receive full credit. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 5 2 5 3 5 4 5 5 5 6 5 TOTAL 30 GOODLUCK * August 11, 2011

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1. Prove that A R n is measurable if and only if A = G \ Z for some G δ -set 1 and some set Z of measure 0 . 2. Show that the inverse function theorem follows from the implicit function theorem. 3. Suppose that U R n is open and that f : U R has continuous second-order partial derivatives on some ball B r ( a ) for some a U , and that f ( a ) = ~ 0 . Suppose that the Hessian of f at a has both positive and negative eigenvalues. Use Taylor’s theorem to prove that f has a saddle point at a . 4. Recall that a point x R is a condensation point of a set A if every neighborhood of x contains uncountably many points in A
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## This note was uploaded on 10/18/2011 for the course MATH S4062Q taught by Professor Staff during the Summer '11 term at Columbia.

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final - Intro to Modern Analysis Summer MMXI Final Exam...

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