Uniform convergence and continuity

[rainwyz0706]

1.kn (x) = 0 for x ≤ n
x − n, x ≥ n,
Is kn(x) uniformly convergent on R?

I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it?

2.Fix 0 < η < 1. Suppose now that h : [0, 1] → R is continuous. Prove that the series
t(x) = ∑ x^n h(x^n ) is uniformly convergent on [0, η]. Deduce that t(x) is continuous.

I'm not sure how to treat h(x^n) here, since it's not bounded. Could anyone help me figure it out?

Any help is appreciated!

 

[lanedance]

for the first, how about a slightly simpler case first... can you show whether y=x is uniformly convergent?

 

[lanedance]

for the 2nd, i would start by considering the definition of uniform convergence

so start with any e>0 and look at |t(x+e) - t(x)|

i haven't tried it yet, but i think the idea is the multiplication of x^n and the continuity of h(x) should do it...

 

[rainwyz0706]

Thanks for your reply. I think clearly y=x is not uniformly convergent, so I guess kn(x) isn't either?
About the second one, I tried to work with the epsilon-delta definition, but the result seemed still depend on n. Could you please be a bit more specific how you would do it?

 

[Tedjn]

Is the first question asking whether k_n(x) is uniformly continuous? If not, is it asking whether the sequence k_n(x) is uniformly convergent to the zero function? For the second question, are you familiar with the theorem that says a continuous function on a compact set achieves its maximum and minimum?

 

[rainwyz0706]

Thanks, I got it!