# Riemann Integral

• January 8th 2012, 04:34 AM
younhock
Riemann Integral
Suppose i have $f(x)=0$ for $x \neq \frac{1}{n},n=1,2,3,.....$ and $f(\frac{1}{n})=1$.
how to show that $f$ is Riemann Integrable and the riemann sum is 0 by using definition?
• January 8th 2012, 05:23 AM
girdav
Re: Riemann Integral
Fix $\varepsilon>0$, and $n_0$ such that $n_0^{-1}\leq \varepsilon$. Put $s_2(x):=\begin{cases} 1&\mbox{ if } x\leq n_0^{-1}\\0&\mbox{ otherwise}\end{cases}$ and $s_1(x)=0$. Choose a good subdivision and you are done.
• January 8th 2012, 08:59 AM
younhock
Re: Riemann Integral
Quote:

Originally Posted by girdav
Fix $\varepsilon>0$, and $n_0$ such that $n_0^{-1}\leq \varepsilon$. Put $s_2(x):=\begin{cases} 1&\mbox{ if } x\leq n_0^{-1}\\0&\mbox{ otherwise}\end{cases}$ and $s_1(x)=0$. Choose a good subdivision and you are done.

Why we construct $s_2 (x)$ and $s_1 (x)?$ Can it related to $f?$
• January 8th 2012, 09:21 AM
girdav
Re: Riemann Integral
It's just a definition. $s_1$ and $s_2$ are step functions. Using the subdivision $0, you can check the definition of Riemann integrable.