Deceptive uniform convergence question

[STEMucator]

Homework Statement

http://gyazo.com/55eaace8994d246974ef750ebeb36069

Homework Equations

Theorem III :
http://gyazo.com/af2dfeb33d3382430d39f275268c15b1

The Attempt at a Solution

At first this question had me jumping to a wrong conclusion.

Upon closer inspection I see the sequence converges to 1 as n goes to infinity for |x|<1. The sequence converges to 0 as n goes to infinity for |x|≥1. Hence the sequence is not uniformly convergent over the whole real line.

If we restrict the domain of x to (-1,1) or (-∞,-1] U [1,∞), then we can observe uniform convergence over each interval respectively.

The question isn't too clear about what it's asking for, but that's my take.

 

[Dick]

Homework Statement

http://gyazo.com/55eaace8994d246974ef750ebeb36069

Homework Equations

Theorem III :
http://gyazo.com/af2dfeb33d3382430d39f275268c15b1

The Attempt at a Solution

At first this question had me jumping to a wrong conclusion.

Upon closer inspection I see the sequence converges to 1 as n goes to infinity for |x|<1. The sequence converges to 0 as n goes to infinity for |x|≥1. Hence the sequence is not uniformly convergent over the whole real line.

If we restrict the domain of x to (-1,1) or (-∞,-1] U [1,∞), then we can observe uniform convergence over each interval respectively.

The question isn't too clear about what it's asking for, but that's my take.

I think you are supposed to conclude that given the theorem, if the $f_n$ are continuous and the limit function is not continuous, then the convergence can't be uniform. And I don't think they converge uniformly on (-1,1) either or any of the other intervals you are talking about. If you think they do please let me know why.

 

[STEMucator]

I think you are supposed to conclude that given the theorem, if the $f_n$ are continuous and the limit function is not continuous, then the convergence can't be uniform. And I don't think they converge uniformly on (-1,1) either or any of the other intervals you are talking about. If you think they do please let me know why.

I see, I think i understand how the theorem and the question relate. I never did check the convergence on the intervals though so I suppose I shouldn't have assumed.