# hw5 - Modern Analysis Homework 5 1 We saw that the implicit...

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Modern Analysis, Homework 5 1. We saw that the implicit function theorem follows from the inverse function theorem. Show that the inverse function theorem follows from the implicit function theorem. 2. Let A,B R n . Set d ( A,B ) = inf {| x - y | ; x A,y B } . If A is compact and B is closed, prove that d ( A,B ) > 0 if A B = {} . 3. 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. 4. Suppose that f : U R n 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 in several variables to prove that f has a saddle point at a . 2 5. (a) Let E R be measurable with μ ( E ) > 0. Show that the set of diﬀerences { d = x - y, x E,y E } contains an interval centered at the origin. ( Hint: Find an open set G containing E such that μ ( G ) < 4
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