18.100pset7

18.100pset7 - 18.100B/C: Fall 2008 Solutions to Homework 7...

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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....
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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.

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18.100pset7 - 18.100B/C: Fall 2008 Solutions to Homework 7...

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