We define a * characteristic function*, , on a set of points on by

Let be a step functions taking the (constant) value on the th open subinterval of a partition of . Prove that for each , we have

* Proof. * First, we note that the open subintervals of some partition of are necessarily disjoint since . Hence, if then for exactly one .

So, we have

for all , and for any . Further, by definition of , we know if . So,

for each

I think the summation result should be c(j) instead of c(k) since you have already let x is an element of I(j)