(1) Let .
(a) Find the pointwise limit of for all
(b) Is the convergence uniform on ?
(c) Is the convergence uniform on ?
(d) Is the convergence uniform on ?
(2) Let .
Find the pointwise limit of on . Is this convergence uniform on ? Is the convergence uniform on all of ?
(3) For each , find the points on where the function attains its maximum and minimum values. Use this to prove converges uniformly on . What is the limit function?
(4) For each , define on by
(a) Find the pointwise limit of on and decide whether or not the convergence is uniform.
(b)Construct an example of a pointwise limit of continuous functions that converges everywhere on the compact set to a limit function that is unbounded on this set.
If anyone can show me how to do these, I would appreciate it, thanks!
That's something I don't understand (someone tried to explain it to me...), when do you know if you fix x or not ? If we can find a sup to the function, which is convergent while n tends to infinity, shouldn't it work ?
The conditions of a uniform convergence seem to apply to n, not to x ?
I'm not sure to understand what you mean. Here we can't fix since we work with . What one sometimes does is showing the uniform convergence on every is that what you mean by "fixing" ?
Yes it should.If we can find a sup to the function, which is convergent while n tends to infinity, shouldn't it work ?
Then you'd rather study ?The conditions of a uniform convergence seem to apply to n, not to x ?
Moo, The sequence of functions in #1 appears in many texts. There is an extensive discussion of this very problem in Advanced Calculus by Taylor & Mann. Because that is such a classical text, any good mathematics library should have a copy.
That sequence converges uniformly on any closed interval that does not contain 0 and does not converge uniformly on any interval if 0 is in its closure.
Hi Plato,
I doubt I can find that easily such a book here
That's what I thought for the convergence, but I didn't see the use of asking for the three intervals... Can you enlighten my blindness here ?
Basically, is there a difference between asking for convergence in and , ?
I don't really remember if one can be considered as a closed interval :/ That's a part I don't like in maths lol.
Thanks:
As uniform convergence on every , does not imply uniform convergence on , fixing is useless here.
Fixing is useful to show the continuity of the limit : if for all , if each is continuous on and if converges uniformly on every (but not necessarily on ) we get that is continuous on every . This does imply its continuity on .