Access over 35 million academic & study documents

Assignment 5 Analysis Mathematics

Content type
User Generated
School
Warwick University Coventry UK
Rating
Showing Page:
1/2
University of Warwick, Coventry, UK
Warwick Mathematics Institute (WMI)
MA244 Analysis III
Assignment 5
1. Sketch the graph of




and find the pointwise
limit


 

For each 

such
that
 

 Deduce that the convergence
is not
uniform.
Show directly that is regulated, and that


(Note that this
does not follow from the theorem about the integral of the limit of a
uniformly convergent sequence of regulated functions.)
2. Let


function (meaning that is differentiable
with
continuous). Define


Show that
converges uniformly (to some limit function to be specified). Show directly
that
[If necessary, think first of a specific example, such as

3. In question 2 show that
uniformly. What difficulty arises with this if
is differentiable but its derivative is not continuous (as, for example,
when



 
4. Let
be given by
 

 Find the inflexion points of
(where

 sketch the graphs of




Find the pointwise limit function 
 

Does
converge pointwise as 
Does
converge uniformly? Does
converge to

Sign up to view the full document!

lock_open Sign Up
Showing Page:
2/2

Sign up to view the full document!

lock_open Sign Up
Unformatted Attachment Preview
University of Warwick, Coventry, UK Warwick Mathematics Institute (WMI) MA244 Analysis III Assignment 5 1 1. Sketch the graph of fn ∶ [−1, 1] → ℝ, fn (x) ∶= x 2n−1 and find the pointwise limit f ∶ [−1, 1] → ℝ, f(x) ∶= lim fn (x). For each n find t n ∈ [−1, 1] such n→∞ that |fn (t n ) − f(t n )| ≥ 1/2. Deduce that the convergence fn → f is not uniform. 1 1 Show directly that f is regulated, and that ∫−1 fn → ∫−1 f. (Note that this does not follow from the theorem about the integral of the limit of a uniformly convergent sequence of regulated fu ...
Purchase document to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Anonymous
Just what I needed…Fantastic!

Studypool
4.7
Indeed
4.5
Sitejabber
4.4