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

natasha d
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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
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You're almost there! You can choose your c_j'' and c_j' to be as close as you want, can't you?
 
make it small enough to neglect the second term on the RHS?
 
Thats the idea. You can choose c_j''-c_j' to be less than some multiple of epsilon and then proceed.
 
got it thanks
00006qsq.jpg
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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