Piecewise Continuity implies Integrability
A piecewise continuous with finite one-sided limits at the partition points is integrable, and .
Proposition 4.13 (Piecewise Continuity Implies Integrability)
If is piecewise continuous, i.e. suppose there is a partition of such that is continuous and has a finite limit as and , for all , then is integrable, and
where
Proof idea
On each subinterval , the function differs from only at the two endpoints, so by the finite-change lemma is integrable and
Each is continuous on a closed bounded interval, hence integrable. Additivity of the integral over adjacent subintervals assembles these pieces into the displayed formula.
Related
Stated in
- Proposition 4.13 (Piecewise Continuity Implies Integrability)§4.2 Integrability Criteria