This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 18.100B/C: Fall 2008 Solutions to Homework 7 1.(a) For a given and x , f ( x, ) is the supremum of a set of positive values. Thus, f ( x, ) > if and only if this set is nonempty. So f ( x, ) > 0 if and only if there exists some > 0 such that for all t X , d X ( x,t ) < implies d Y ( f ( x ) ,f ( t )) < . That this statement is true for every > is exactly the definition of continuity of f at x , so were done. (b) First suppose that f is unformly continuous and choose > 0. Then there exists some > 0 such that for any x,y X we have d Y ( f ( x ) ,f ( y )) < whenever d X ( x,y ) < . It follows that f ( x, ) for every x X , so inf x X f ( x, ) > 0. Conversely, suppose inf x X f ( x, ) > 0, and define = 1 2 inf x X f ( x, ). It follows that < < f ( x, ) for every x X . By the properties of the supremum, this means that for each x X there exists some c > such that c belongs to the set of which f ( x, ) is the supremum, i.e. d X ( x,t ) < c implies d Y ( f ( x ) ,f ( t )) < for every t X . Since < c we have that { t X  d X ( x,t ) < } { t X  d X ( x,t ) < c } and thus d X ( x,t ) < implies d Y ( f ( x ) ,f ( t )) < for all x,t X . But this is exactly the definition of uniform continuity for f , so were done. (We introduced a factor of 1 2 in this last paragraph not suggested by the hint. It may be possible to set = inf x X f ( x, ) directly, without the 1 2 factor, and still produce a valid proof. The one problem to watch out for is the following situation: for fixed > 0, what if f ( x, ) = d is constant but d negationslash { >  t Xd X ( x,t ) < = d Y ( f ( x ) ,f ( t )) < } for some x (i.e., the supremum is not achieved)?) (c) For any x,t X we have  f ( x ) f ( t )  =  x 2 t 2  =  x t   x + t  < (2 x + ) , so 2 x + 2 is an upper bound for { f ( x ) f ( t )   t X,d X ( x,t ) < } . It addition, the limit as t approaches x + from below of  f ( x ) f ( t )  is equal to (2 x + ), so it is in fact the least upper bound, as desired....
View
Full
Document
This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
 Fall '10
 Prof.KatrinWehrheim

Click to edit the document details