18 100B Fall 2002 Homework 6 Due by Noon Tuesday October 29 Rudin 1 Chapter 4 Problem 20 If E is a nonempty subset of a metric space X de ne the distance from x X to E by E x inf d x z z E a Prove that E x 0 if and only if x E b Prove that E is uniformly continuous on X by showing that E x E y d x y for all x y X Solution a If E x 0 then there exists a sequence zn E such that Conversely if d x zn 0 This implies zn x and hence x E x E then either x E in which case E x 0 or else x E so there exists a sequence zn E with zn x This implies d x zn 0 so E x 0 b If x y X then for any z E using the triangle inequality E x d x z d x y d y z Taking the in mum over z E on the right hand side shows that E x E y d x y Interchanging the roles of x and y gives the desired estimate E x E y d x y This proves the uniform continuity of E since given 0 d x y implies E x E y 2 Chapter 4 Problem 23 A real valued function de ned on a b is said to be convex if f x 1 y f x 1 f y whenever x y a b and 0 1 Prove that every convex function is continuous Prove that every increasing convex function of a convex function is convex If f is convex on a b and if a s t u b show that f t f s f u f s f u f t t s u s u t Solution c We do the third part rst Since a s t u b u t t s 1 u with u s 0 1 Thus f t 1 u t t s f s f u t s f u u t f s u s f t 0 u s u s This can be rewritten as t s f u f s u s f t f s 0 and u s f u f t u t f u f s 0 proving the two desired inequalities f t f s f u f s f u f t t s u s u t 1 2 a Given x a b choose 0 so that x x a b Now consider a point z x x applying the second inequality in 1 gives the rst inequality in 2 f x f x f z f x f x f x z x Applying the outer inequality in 1 to the three points z x x gives the second inequality Now consider the case x z x Then the rst inequality in 2 follows from the outer inequality in 1 applied to the three points x x z and the second inequality in 2 follows from the rst inequality in 1 applied to x z x Now 2 implies that f x f z C x z z x x and hence proves the continuity of f in fact the Lipschitz continuity c Let g be convex and increasing on c d and f be convex on a b with f a b c d Then set h x g f x Since f is convex A f x 1 y f x 1 f y B and since g is increasing g A g B so h x 1 y g f x 1 y g f x 1 f y h x 1 h y proving the convexity of h 3 Chapter 4 Problem 26 Suppose X Y and Z are metric spaces and Y is compact Let f X Y and let g Y Z be continuous and 1 1 and put h x g f x Prove that f is uniformly continuous if h is uniformly continuous Show that compactness of Y cannot be omitted from the hypotheses even when X and Z are compact Solution Consider the subset Z g Y as a metric space with the metric induced from Z Then g Y Z is 1 1 and onto Since Y is compact so is Z and by a result from class the inverse of g is continuous Thus again by a result from class both g and g 1 Z Y are uniformly continuous Note that the composite of two uniformly continuous maps is uniformly continuous1 Applying this to f g 1 h h g f shows that the uniform continuity of h implies that of f As a counterexample to the result when the compactness of Y is dropped take X Z 0 1 and Y 0 12 1 32 Let f be the discontinuous map f x x for 0 x 12 f x x 12 for 12 x 1 Then let g be the continuous map g y y for 0 y 12 and g y y 12 for 1 y 32 Observe that g is uniformly continuous since g y g y y y The composite map is the identity on 0 1 so uniformly continuous but f is not even continuous of course if it was continuous it would be uniformly continuous since 0 1 is compact 1If the maps are f X Y and g Y Z both uniformly continuous then given 0 there exists 0 sucht that dY y y implies dZ g y g y Then from the uniform continuity of f there exists 0 such that dX x x impies dY f x f x and hence d g f x g f x But this is the uniform continuity of h g f 3 4 Chapter 5 Problem 1 Let f be de ned for all real x and suppose that f x f y x y 2 x u R Prove that f is constant Solution Certainly f is di erentiable at each point with derivative zero since f x h f x lim lim h 0 0 h 0 0 h 0 h By the mean value theorem it follows that f is constant 5 Chapter 5 Problem 2 Suppose f x 0 in a b Prove that f is strictly increasing in a b and let g be its inverse function Prove that g is di erentiable and 1 g f x x a b f x Proof By the mean value theorem if y x are two points in a b then there exists z x y such that f y f x y x f z 0 Thus f is stricly increasing It follows that it is 1 1 as a map onto the possibly in nite interval c d inf f sup f Thus it has an inverse g determined by the fact that g y x if f x y