Let >0. Put M=sup f
.Now choose a partition so that the total length of the intervals containing the discontnuities of f
f is smaller than
The contribution from the intervals with discontinuities of f
f is smaller than
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
Assemble the pieces and you have Riemann integrability.
so f is Riemann integral.
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