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Unformatted text preview: Mathematics 250 Fall 2008 Proof Practice # 2 a) Let F : D R m be a vector-valued function defined on an open domain D R n . Lets call F locally Lipschitz if, given any point x in D , we can find a ball B r ( x ) around x , such that F is Lipschitz on B r ( x ). (Note that here r depends on x , and the Lipschitz constant of F on B r ( x ) depends on both x and r .) Argue that, if F is locally Lipschitz on D , then F is continuous on D . b) Consider the mappings A and M from R 2 to R defined by addition and multiplication: A ( x y ) = x + y, and M ( x y ) = xy. Show that A is Lipschitz, and that M is locally Lipschitz. Is M (globally) Lipschitz? Response: a) Assume that F is locally Lipschitz. Let x be any point in D . We want show that F is continuous at x . To do this, we must show that, given > 0, we can find > 0 such that, if || z- x || < , then || F ( z )- F ( x ) || < ....
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- Fall '06