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 Homework 7 Available Monday, October 20 Due Wednesday, October 29 1. Let f : X Y be a continuous function between metric spaces. Define a function f : X (0 , ) (0 , ) as follows: f ( x, ) = sup { > 0 ; t X d X ( x, t ) < d Y ( f ( x ) , f ( t )) < } . (a) (5pts) Show that the statement f is continuous at x is equivalent to f ( x, ) > for each > . (b) (5pts) Show that f is uniformly continuous on X iff inf x X f ( x, ) > . [ Hint : you need to find a uniform > ; choose it to be this infimum.] (c) (5pts) Consider the function f ( x ) = x 2 defined on the metric space X = [0 , ) . Show that, for each x X , sup t X,d X ( x,t ) <  f ( x ) f ( t )  = 2 x + 2 . (d) (5pts) Use part (c) to show that f ( x, ) = x 2 + x . Show that, for fixed > , lim x [ x 2 + x ] = 0 . Conclude that f is not uniformly continuous on X . [ Hint : you can use Calculus here, but you neednt. Setyou can use Calculus here, but you neednt....
View
Full
Document
 Fall '10
 Prof.KatrinWehrheim

Click to edit the document details