Thread: proving function is riemann integrable

1. proving function is riemann integrable

the value of a and b is not given. I don't understand how to prove.

2. Originally Posted by xyz

the value of a and b is not given. I don't understand how to prove.

Here is a hint. How would you do it for one point?

i.e let $c \in (a,b)$

$f(x)=\begin{cases} 0, x \ne c \\ M, x=c \end{cases}$

$\int_{a}^{b}f(x)dx$

Let P be the partition $\{a,c-\frac{\epsilon}{2|M|},c+\frac{\epsilon}{2|M|},b\}$

Now how would you do it for 2,3 or any finite number?

I hope this helps.

3. Originally Posted by xyz
the value of a and b is not given. I don't understand how to prove.
Do you know that B Riemann was a real person?
So use capital letters. It is the Riemann integral.

Can you find a partition of $[a,b]$ that has each $x_j$ as an endpoint?
If so, what are the upper and lower sums?

4. I understand the strategy. The textbook gives 2 hints. It says to use theorem 3.1.2 and to use the solution of the previous problem. Unfortunately, I have not been able to find the solution of previous problem. I posted the previous problem at

http://www.mathhelpforum.com/math-he...ntegrable.html

but no one gave any idea. If I get an idea out of the previous problem, I can solve this problem because it appears to be related.

Partition of [a, b] will be (b-a) / n where n is the number of increments. So the partition will look like

{a, a + (b-a)/n , a + (b-a)/n + (b-a)/n, ......b}