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Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E R, for each n e Z+ (i) Show that the sequence {fn}-1 converges uniformly to f in R. (ii) Show that periodicity is necessary: Give an example of a continuous function f : R→R such that the sequence f„(x) := f(x+-) does not converge to f uniformly.

Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E R, for each n e Z+ (i) Show that the sequence {fn}-1 converges uniformly to f in R. (ii) Show that periodicity is necessary: Give an example of a continuous function f : R→R such that the sequence f„(x) := f(x+-) does not converge to f uniformly.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 51E
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Suppose f : R → R is a continuous, 2n – periodic function. Define the function f,(x) := f(x + 1) for x E R, for each n E Z- (i) Show that the sequence {f,}=1 converges uniformly to f in R. (ii) Show that periodicity is necessary: Give an example of a continuous function ƒ : R→R such that the sequence f,(x) := f(x+-) does not converge to f uniformly.
Transcribed Image Text:Suppose f : R → R is a continuous, 2n – periodic function. Define the function f,(x) := f(x + 1) for x E R, for each n E Z- (i) Show that the sequence {f,}=1 converges uniformly to f in R. (ii) Show that periodicity is necessary: Give an example of a continuous function ƒ : R→R such that the sequence f,(x) := f(x+-) does not converge to f uniformly.
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