Re: uniform convergence?


damage wrote:
> i know the function f_n:[-1,1]-->reals where f_n(x)=x^(1/(2*n-1)) converges
> pointwisesly to -1 (for -1</= x <0) and 1 for x>/= 0
> what i want to know is when should a cts function converge to a limit which
> is cts, is my thinking right when i say.
> the reason this does not converge to a cts function (every though it is a
> family of cts fctns) is due to the fact that the derivative tends to
> infinity (f '_n(x)).
> my question is ..does uniform convergence => derivatives converge? if so
> could someone point out a theorem? (web search didnt come up with anything
> helpful)
> if this is not the case, is there a counter example (i dont believe this
> case actually exists though!)
>
> Many thanks

Rudin's analysis book Prinicples of Mathematical Analysis has a chapter
(chapter 7) devoted to series and sequences of functions. He answers
the kind of the questions you're asking, as well as many others. It's
a little expensive to buy, but might be worth checking out from the
library.

Anyway, to answer one of your questions, a uniformly convergent
sequence of continuous functions converges to a continuous function.
In your case, f_n(x) does converge uniformly if restricted to certain
intervals, but does not converge uniformly on [-1, 1].

I could quote you more results from Rudin about sequences of functions,
but if you're interested I would suggest just checking it out.

.