Function bounded on [a,b] with finite discontinuities is Riemann integrable

[natasha d]

Homework Statement

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

Homework 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)<$\epsilon$

The Attempt at a Solution

the term on the LHS must be made <ε

 

[Oster]

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

 

[natasha d]

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

 

[Oster]

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

 

[natasha d]

got it thanks