From the definition, we know that if exist and have same upper bound and lower bound when the selecting point changing.Coz there will be some bad subinterval happen, but we know the number of it is countable. It does not matter after we divide the function into n which is large enough parts. The upper bound will become smaller and smaller, and the lower bound will become bigger and bigger until they become the same.Which means if the removed points is not infinity , the function still integrable.