Question

If f is continuous except at finitely many points in I,then f is Riemann-integrable.

Answer 1

Let $\isin$ >0. Put M=sup$_{x\in[a,b]}$ f

.Now choose a partition so that the total length of the intervals containing the discontnuities of f

f is smaller than $\dfrac{\in}{2M}$

The contribution from the intervals with discontinuities of f

f is smaller than $\dfrac{\in}{2}$

Since f is continuous on the rest, it is fairly easy to engineer an argument that supplies a partition whose upper and lower sums differ by less than $\dfrac{\in}{2}$

Assemble the pieces and you have Riemann integrability.

so f is Riemann integral.