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18.100B.Homework6

# 18.100B.Homework6 - 18.100B/C Fall 2008 Homework 6...

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18.100B/C: Fall 2008 Homework 6 Available Wednesday, October 15 Due Wednesday, October 22 1. (10pts) Problem 1, page 98 in Rudin 2. (10pts) Problem 3, page 98 in Rudin 3. (10 pts) Let ( X, d ) be a metric space. Fix x 0 X and a continuous function g : R R . Show that the function X R deﬁned by x ±→ g ( d ( x, x 0 )) is continuous. 4. Let ( S, d S ) be a set equipped with the discrete metric (i.e. d S ( t, r ) = 1 for t ² = r ). (a) (5pts) Show that any map f : S X into another metric space X is continuous; using the deﬁnition of continuity by sequences. (b) (5pts) Show that any map f : S X into another metric space X is continuous; using the deﬁnition of continuity by ± - and δ -balls. (c) (10pts) Which maps f : R S are continuous? (Give an easy characterization and prove it.) 5. Consider the function h : Q R given by h ( x ) = ( 0 ; x 2 < 2 , 1 ; x 2 > 2 . (a) (5pts) Is h continuous? (a)

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18.100B.Homework6 - 18.100B/C Fall 2008 Homework 6...

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