Sequences and Series of Functions Question (Rudin Chapter 7)

[gajohnson]

Homework Statement

The problem is Exercise 8 from Chapter 7 of Rudin. It can be seen here:

http://grab.by/mGxY

Homework Equations

The Attempt at a Solution

It seems quite obvious to see that because [itex]\sum\left|c_n\right|[/itex] converges, [itex]f(x)[/itex] will converge uniformly.

However, I am having a difficult time understanding why [itex]f(x)[/itex] will be continuous at all [itex]x\neq{x_n}[/itex]

Any help with understanding this second part of the proof would be greatly appreciated. Thanks!

 

[gajohnson]

Is this as simple as stating that there is a jump discontinuity at [itex]x=x_n[/itex], because the left-hand limit = 0, and the right-hand limit = 1?

 

[verty]

Which c_n's does I(x - x_n) choose to exclude from the sum?

 

[gajohnson]

Which c_n's does I(x - x_n) choose to exclude from the sum?

I'm not quite sure I understand your question. Do you mean for me to say that [itex]c_n[/itex] is excluded from the sum when [itex]I(x-x_n)≤0[/itex]?

 

[verty]

I'm not quite sure I understand your question. Do you mean for me to say that [itex]c_n[/itex] is excluded from the sum when [itex]I(x-x_n)≤0[/itex]?

I can't really say more without being too helpful.