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non uniform convergent sequence of functions

Posted July 4th, 2010 by Structure
$$\displaystyle \mathop{\mathrm{f_n}}(x)=\displaystyle \left\{ \begin {array}{ll} nx & \mbox{when}\,\, 0\le x\le \frac{1}{n}, \\2-nx & \mbox{when}\,\, \frac{1}{n}<x<\frac{2}{n},\\ 0 & \mbox{when}\,\, \frac{2}{n}\le x\le 1. \\ \end{array} \right.$$

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

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

so

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

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