# Homework Help: Function bounded on [a,b] with finite discontinuities is Riemann integrable

1. Mar 14, 2012

### natasha d

1. The problem statement, all variables and given/known data

to prove that a function bounded on [a,b] with finite discontinuities is Riemann integrable on [a,b]

2. Relevant equations

if f is R-integrable on [a,b], then $\forall$ $\epsilon$ > 0 $\exists$ a partition P of [a,b] such that U(P,f)-L(P,f)<$\epsilon$

3. The attempt at a solution
the term on the LHS must be made <ε

Last edited: Mar 14, 2012
2. Mar 14, 2012

### Oster

You're almost there! You can choose your c_j'' and c_j' to be as close as you want, can't you?

3. Mar 14, 2012

### natasha d

make it small enough to neglect the second term on the RHS?

4. Mar 14, 2012

### Oster

Thats the idea. You can choose c_j''-c_j' to be less than some multiple of epsilon and then proceed.

5. Mar 14, 2012

### natasha d

got it thanks