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Analysis convergence of sequences of functions

AJC
Post Icon Posted: Submitted by AJC on 8 April 2010 - 4:18pm.

Joined: 2010-04-04
Posts: 3

If a sequence of functions is uniformly convergent to a function f then it is pointwise convergence, but if it is pointwise convergent it is not neccessarily uniformly convergent.

Is there a classic example of a sequence of functions that is pointwise convergent but not uniform, as I can't seem to imagine one?

‹ 2nd yr algebra I 2009paper last question???? Measure Theory Question ›
Matthew
Post Icon Posted: 12 April 2010 - 12:26am

Joined: 2007-11-03
Posts: 12

How about $ f_n(x)=1 $ on $ (n,n+1) $ and $ f_n(x)=0 $ everywhere else. Then $ f_n $ converges pointwise to $ f=0 $ but $ \forall n \exists x f_n(x)=1 $. There are plenty more similar examples, see if you can find a similar sequence of functions on a bounded open interval.

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