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.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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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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