# Uniform Convergence

• Oct 19th 2011, 02:56 AM
hmmmm
Uniform Convergence
Am I right in saying that if I have $\displaystyle f_n=\frac{x^n}{x^n+1}$ and

$\displaystyle f(x) = \left\{\begin{array}{c l} 0 & x \in [0,1)\\ 0.5 & x =1 \\ 1 & x\in(1,\infty)\end{array}\right.$

definied on $\displaystyle (0,\infty)$ is not uniform convergent to f(x) (the problem being near the 1)?

Thanks for any help
• Oct 19th 2011, 03:08 AM
emakarov
Re: Uniform Convergence
According to the uniform limit theorem, the uniform limit of a sequence of continuous functions is continuous, so you are right.
• Oct 19th 2011, 03:13 AM
hmmmm
Re: Uniform Convergence
Cool thanks (I have proved it from the definition, which is a lot longer, feel a bit foolish now for asking!) it is pointwise convergent to f(x) though right?
• Oct 19th 2011, 03:18 AM
emakarov
Re: Uniform Convergence
Yes.

Hint: To avoid <br/> in LaTeX formulas, remove all line breaks between $$and$$ (not LaTeX line breaks \\; just put the whole formula on one editor line).
• Oct 19th 2011, 03:23 AM
hmmmm
Re: Uniform Convergence
Thanks very much for all the help (sorry about the poor LaTex and thanks for editing it)
• Oct 19th 2011, 03:25 AM
emakarov
Re: Uniform Convergence
Ha, it was Plato who edited it!