M25008PP2

M25008PP2 - Mathematics 250 Fall 2008 Proof Practice # 2 a)...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ) || < ....
View Full Document

Ask a homework question - tutors are online