# riemann integration help

• Dec 5th 2009, 07:54 PM
binkypoo
riemann integration help
I need help with the following:
suppose that f:[0,1]-->R is defined by:
f(x)=1/n if 1/(n+1) < x <= 1/n
and 0 if x=0.
Verify that f is riemann integrable on the interval [0,1]
• Dec 6th 2009, 12:45 AM
tonio
Quote:

Originally Posted by binkypoo
I need help with the following:
suppose that f:[0,1]-->R is defined by:
f(x)=1/n if 1/(n+1) < x <= 1/n
and 0 if x=0.
Verify that f is riemann integrable on the interval [0,1]

According to your definition, $\displaystyle f(x)$ is only defined on the set $\displaystyle \left(\frac{1}{n+1},\frac{1}{n}\right)\cup\{0\}$ and not on $\displaystyle [0,1]$ . Check this.

Tonio
• Dec 6th 2009, 01:53 PM
binkypoo
please excuse me, I should have used latex,
$\displaystyle x \leq 1/n$
sorry.
• Dec 6th 2009, 01:59 PM
tonio
Quote:

Originally Posted by binkypoo
please excuse me, I should have used latex,
$\displaystyle x \leq 1/n$
sorry.

I don't get it: what is then the function's definition?

Tonio
• Dec 6th 2009, 02:58 PM
binkypoo
$\displaystyle f(x)=\left\{ \begin{array}{lr} \frac{1}{n} & : \frac{1}{n+1}< x\leq \frac{1}{n}\\ 0 & : x=0 \end{array}\right.$
• Dec 6th 2009, 02:59 PM
Plato
Quote:

Originally Posted by tonio
I don't get it: what is then the function's definition?

The collection of intervals $\displaystyle \left\{ {\left( {\frac{1}{{n + 1}},\frac{1}{n}} \right]:n \in \mathbb{Z}^ + } \right\}$ partitions $\displaystyle (0,1]$.

So the function is $\displaystyle f:[0,1]\mapsto [0,1]$ by $\displaystyle f(x) = \left\{ {\begin{array}{rl} 0 & {,x = 0} \\ {\frac{1}{n}} & {,x \in \left( {\frac{1}{{n + 1}},\frac{1}{n}} \right]} \\ \end{array} } \right.$