let f be uniformly continuous on R and define a sequence of functions by Fn(x) =

ID: 3081614 • Letter: L

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let f be uniformly continuous on R and define a sequence of functions by Fn(x) = F(x + (1/n)). a. show that Fn --> F uniformly on R. b. propose an example to show that (a) fails when F is only continuous but bot uniformly continuous on R.

Explanation / Answer

sequence of functions {f_n} is said to converge uniformly to a function f, if for every e > 0, there is a natural number N, such that for all x, and for all n >=N then |f_n(x) - f(x)| 0. Let N be a natural number such that N > 1/e. (This is possible since we can always find a natural number bigger than any given real number). Hence 1/N =N that |f_n(x) - f(x)| = N. Then 1/m

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let f be uniformly continuous on R and define a sequence of functions by Fn(x) =

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