SUMMARY
A function that is bounded on the interval [a,b] and has a finite number of discontinuities is Riemann integrable on that interval. The key to proving this is to utilize the definition of Riemann integrability, which states that for any ε > 0, there exists a partition P of [a,b] such that the difference between the upper sum U(P,f) and the lower sum L(P,f) is less than ε. By carefully selecting the points c_j'' and c_j' within the partitions, one can ensure that the difference can be made arbitrarily small, thus confirming Riemann integrability.
PREREQUISITES
- Understanding of Riemann integrability and its definition
- Familiarity with the concepts of upper and lower sums in integration
- Knowledge of partitioning intervals in calculus
- Basic proficiency in handling ε-δ arguments in mathematical proofs
NEXT STEPS
- Study the properties of Riemann integrable functions
- Learn about the implications of the Heine-Cantor theorem on bounded functions
- Explore examples of functions with finite discontinuities and their integrability
- Investigate the relationship between Riemann and Lebesgue integrability
USEFUL FOR
Students and educators in calculus, mathematicians interested in real analysis, and anyone studying the properties of integrable functions.