A series of functions that converges pointwise

[R.P.F.]

Homework Statement

Hi,

Can someone give me an example where f_n is continuous on [0,1] for each n.
f = \sum_{n=0}^\infty f_n 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.

 

[radou]

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.

 

[R.P.F.]

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.

 

[radou]

Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series f_{n}(x)=\frac{x^2}{(1+x^2)^n}, on an interval containing 0.

 

[R.P.F.]

Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series f_{n}(x)=\frac{x^2}{(1+x^2)^n}, on an interval containing 0.

I see. Thanks!