Definition (Definite Integral): Let be continuous on the closed interval

Integrability Theorem: Let f be piecewise continuous ⁵ on . Then f is Riemann/Darboux integrable over ; that is, exists. In particular, if f is continuous on , f is integrable over .
Using the Integrability Theorem, we clearly have the following theorem.
Riemann Regular Integrability Theorem: Let f be continuous on and let be the regular partition that corresponds to , where Then, for any choice of ,

Some conditions that are equivalent to integrability are given in the following theorem.