non uniform convergent sequence of functions

Submitted by Structure on Sun, 07/04/2010 - 16:52.

$\displaystyle \mathop{\mathrm{f_n}}(x)=\displaystyle \left\{ \begin {array}{ll} nx & \mbox{when}\,\, 0\le x\le \frac{1}{n}, \\2-nx & \mbox{when}\,\, \frac{1}{n}<x<\frac{2}{n},\\ 0 & \mbox{when}\,\, \frac{2}{n}\le x\le 1. \\ \end{array} \right.$

This is a sequence of continuous functions.
We have for any $x\in [0,1]$

$\lim_{n\to \infty}f_n(x)=0$

We also have

$\int_0^1f_n(x)dx=1$

but

$\int_0^1f(x)dx=0$

$\lim_{n\to \infty}\int_0^1f_n(x)dx\ne\int_0^1\lim_{n\to\infty}f_n(x)dx$

Navigation

User login

non uniform convergent sequence of functions