18.100B.Homework7

18.100B.Homework7 - 18.100B/C: Fall 2008 Homework 7...

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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....
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18.100B.Homework7 - 18.100B/C: Fall 2008 Homework 7...

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