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A series of functions that converges pointwise

  • Thread starter R.P.F.
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  • #1
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Homework Statement



Hi,

Can someone give me an example where f_n is continuous on [0,1] for each n.
[tex]f = \sum_{n=0}^\infty f_n[/tex] converges pointwise(not uniformly) on [0,1] and f is not continuous on [0,1]?

Thanks!

Homework Equations





The Attempt at a Solution



Technically we need uniform convergence, but I am having trouble coming up with such an example.
 

Answers and Replies

  • #2
radou
Homework Helper
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A typical example is the sequence of functions fn(x) = x^n on [0, 1]. It converges pointwise, but not uniformly. And its limit f is not continuous.
 
  • #3
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A typical example is the sequence of functions fn(x) = x^n on [0, 1]. It converges pointwise, but not uniformly. And its limit f is not continuous.
Hey sorry not to be clear. I meant that the infinite sum converges. So I'm talking about a series of functions, not a sequence of functions.
 
  • #4
radou
Homework Helper
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Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series [tex]f_{n}(x)=\frac{x^2}{(1+x^2)^n}[/tex], on an interval containing 0.
 
  • #5
212
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Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series [tex]f_{n}(x)=\frac{x^2}{(1+x^2)^n}[/tex], on an interval containing 0.
I see. Thanks!
 

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