Show the details please. 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 suchthat the sequence f,(x) := f(x+-) does not converge to f uniformly.

Name: Show the details please. Suppose f : R →R is a continuous, 2n – period
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Show the details please. 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:

Given Information - f is a continuous function on R and is 2π periodic.

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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.