The FTOC states that if is continuous on then it is integrable.
If is not defined at certain points of we can often give meaning to an improper integral. But under what circumstances will always be integrable on it's domain, properly or improperly?
If we partition and consider on the subintervals and assume it is continuous, (so is piecewise continuous on ), is then integrable on the entire interval simply by the adding up each integral over each subinterval? But the interval is open, so how does FTOC and improper integrals come into play? Will the improper integral over always be well-defined?