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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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This document was uploaded on 10/26/2011 for the course MATH 18.100B at MIT.
 Fall '10
 Prof.KatrinWehrheim
 Continuity

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