MIT18_100BF10_pset7

MIT18_100BF10_pset7 - 1 ; x 2 > 2 . (a) Is h continuous?...

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± 18.100B : Fall 2010 : Section R2 Homework 7 Due Tuesday, October 26, 1pm Reading: Tue Oct.19 : continuity, Rudin 4.1-12 Thu Oct.21 : Quiz 2 (covering Rudin sections 2.45-47 and 3), p spaces. 1 . (a) Problem 1, page 98 in Rudin (b) Problem 3, page 98 in Rudin 2 . 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 m→ g ( d ( x, x 0 )) is continuous. 3 . Let ( S, d S ) be a set equipped with the discrete metric (i.e. d S ( t, r ) = 1 for t = n r ). (a) Show that any map f : S X into another metric space X is continuous; using the definition of continuity by sequences. (b) Show that any map f : S X into another metric space X is continuous; using the definition of continuity by ǫ - and δ -balls. (c) Which maps f : R S are continuous? (Give an easy characterization and prove it.) 4 . Consider the function h : Q R given by 0 ; x 2 < 2 , h ( x ) =
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Unformatted text preview: 1 ; x 2 > 2 . (a) Is h continuous? (b) Can h be continuously extended to h ˜ : R → R ? (I.e. such that h ˜ ( x ) = h ( x ) for all x ∈ Q .) z ∞ 1 n 5 . Prove that the function f : C → C , z m→ e = n =0 n ! z is continuous by following the steps below. (a) Fix T > and ε > . Show that there exists an N ∈ N such that ∞ ² t n < ε ∀ t ∈ [0 , T ] . n ! n = N [ Hint : The series for e T converges.] (b) Show continuity at z ∈ C by splitting N − 1 ∞ ∞ ² 1 ² 1 ² 1 e z − e x = ( z n − x n ) + z n − x n . n ! n ! n ! n =1 n = N n = N [ Hint : First use (a) with T = | z | +1 , then use the fact that polynomials are continuous.] MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT18_100BF10_pset7 - 1 ; x 2 > 2 . (a) Is h continuous?...

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