Def Def 28.1 π (π,π)=ββπβπΞππ(ππ) supπ π(π,π)=βππ Where β is in Darboux upper sum Def 30.1 limπβΆβ1πβππ=1π(π+ππβππ)=β«πππ(π₯)ππ₯ Or limπβββπβ1π=0π(π₯π)(π₯π+1βπ₯π)=β«πππ(π₯)ππ₯ Basically we just need the parition be small enough. Lemma 30.2 π (π+ππ,π)=π (π,π)+ππ (π,π) for [π,π]=[π,π]βͺ[π,π] π=π[π,π]βͺπ[π,π] where itβs partition Ξπ[π,π]βͺ{π}+Ξπ[π,π]βͺ{π}β€2Ξπ Def 10.2 Ξπ(π)=β(π₯π+1βπ₯π)(supπβinfπ)