Thread: Integrating Functions with Discontinuities

1. Integrating Functions with Discontinuities

Hey, I've got a real analysis question here. Any help would be greatly appreciated:

Let f(x) = {1 if x = 1/n for some n Є N
{0 otherwise

Show that f is integrable on [0,1] and compute
$\displaystyle \int_1^0 {f}$

2. Originally Posted by ajj86
Hey, I've got a real analysis question here. Any help would be greatly appreciated:

Let f(x) = {1 if x = 1/n for some n Є N
{0 otherwise

Show that f is integrable on [0,1] and compute
$\displaystyle \int_1^0 {f}$

consider this partition let $\displaystyle \epsilon >0$

P\fix N such that $\displaystyle \frac{1}{N} < \frac{\epsilon}{4}$

$\displaystyle p_n=\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}},\frac{1}{n}+\frac{\epsilo n}{2^{n+1}} \right)$

for n=2 to n=N

notice that $\displaystyle \sum_{n=2}^{N}\left(\frac{1}{n}+\frac{\epsilon}{2^ {n+1}}-\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}}\right) \right)=\sum_{n=2}^{N}\frac{\epsilon}{2^{n}}<\frac {\epsilon}{2}$

See if you can put it together from here

Good luck.