6. Let X be a metric space. Let f: X → R and g : X - R be real-valued Lipschitz functions with Lipschitz constants K and M, respectively. (a) Prove that f +g is a Lipschitz function. (b) Prove that if f and g are bounded, then f g is a Lipschitz function. (c) Show that the boundedness condition in part (b) is necessary. Do so by giving an example of two Lipschitz functions whose product is not a Lipschitz function.
6. Let X be a metric space. Let f: X → R and g : X - R be real-valued Lipschitz functions with Lipschitz constants K and M, respectively. (a) Prove that f +g is a Lipschitz function. (b) Prove that if f and g are bounded, then f g is a Lipschitz function. (c) Show that the boundedness condition in part (b) is necessary. Do so by giving an example of two Lipschitz functions whose product is not a Lipschitz function.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 20E: Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ]...
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