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Intro to Analysis in-class_Part_52

# Intro to Analysis in-class_Part_52 - 7.3 Uniform...

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7.3 Uniform convergence 109 Example 7.3.2. Let f n ( x ) = x n on I = [0 , 1]. Then { f n } converges pointwise and f ( x ) = 0 , 0 x < 1 , 1 , x = 1 . So a sequence of continuous functions can converge pointwise to something which is not continuous! In fact, f n C ( I ), but f / C ( I )! Even worse: Example 7.3.3. Let f k ( x ) = lim n →∞ (cos k ! ) 2 n . Then whenever k ! x is an integer, f k ( x ) = 1. If x = p/q is rational, then for k q , f n ( x ) = 1. If k ! x is not an integer (for example, if x is irrational), then f k ( x ) = 0. We have an everywhere discontinuous limit function f ( x ) = lim k →∞ lim n →∞ (cos k ! ) 2 n = 0 , x R \ Q , 1 , x Q . Example 7.3.4. Let f n ( x ) = x 2 (1+ x 2 ) n on R and consider f ( x ) = X n =0 f n ( x ) = X n =0 x 2 (1 + x 2 ) n = 0 , x = 0 , 1 + x 2 , x 6 = 0 , since the series is geometric for x 6 = 0. So a series of continuous functions can converge pointwise to something which is not continuous! (Not even integrable!) Example 7.3.5. Let f n ( x ) = sin nx n on R . Then f ( x ) = lim n →∞ f n ( x ) = 0 , x R , so f 0 ( x ) = 0. On the other hand,

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