# Assignment 5 Analysis Mathematics

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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

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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 ...
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