Let f : [0, 1] → R be a continuous function. Show that the sequence of functions1fn(x)(f(х +п -1+ f(-„x + 1+ f(=+))nis uniformly convergent on [0, 1]. What is its limit function?

Answered: Image /qna-images/answer/e9ec3bd0-20fe-451a-94ff-e27b32a59e94.jpg

Answered: Let f : [0, 1] → R be a continuous…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

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...

See similar textbooks

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
Is Uniform convergence of {fn(x)} is sufficient but not necessary to transmit continuity from the individual terms to the limit function ?
Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?
Suppose f is continuous on [0, ∞) and limx→∞ f (x) = 1 . Is itpossible that ∫ 0∞f (x) dx is convergent?
Suppose that fn : [0, 1] → R is defined by fn(x) = x n. If 0 ≤ x < 1, then xn → 0 as n → ∞, while if x = 1, then x n → 1 as n → ∞. So fn → f pointwise where Although each fn is continuous on [0, 1], their pointwise limit f is not (it is discontinuous at 1). Thus, pointwise convergence does not, in general, preserve continuity.
Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].
Let (a) Find the pointwise limit of (fn) for all x ∈ (0,∞).(b) Is the convergence uniform on (0,∞)?(c) Is the convergence uniform on (0, 1)?(d) Is the convergence uniform on (1,∞)?
Suppose f is continuous on [0 , infinity) and limit x appraoaches infinity f(x) =1. Is itpossible that integral 0 to infinity f(x) dx is convergent
Let f_n(x)= nx / 1+nx^2. Find the pointwise limit of (f_n) for all x in (0,infinity). Is the convergence uniform on (0,inifinity)? Is the convergence uniform on (1,infinity)?
Let fn(x) = x^n for x ∈ [0,1]. check if it is pointwise convergence. Define where it becomes discontinuous.
Is there a sequence of continuous functions: fn : [0,1]-->R that is pointwise convergentbut the limit function is discontinuous at each point in [0, 1]?
Suppose that F(u) denotes the DFT of the sequence of f(x)={1, 2, 3, 4}? What is the value of F(14)? (Hint: DFT periodicity)

SEE MORE QUESTIONS

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

SEE MORE TEXTBOOKS

Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn (x) = ,x +1 + f( +...+ f(-T x + n – 1 n n is uniformly convergent on [0, 1]. What is its limit function?