A function can have indefinite integral, but not definite integral; There can also be definite integral, but there is no indefinite integral. A continuous function must have definite integral and indefinite integral; If there are only a finite number of discontinuous points, the definite integral exists; If there is jump discontinuity, the original function must not exist, that is, the indefinite integral must not exist.

Main advantages:

For a function f, if the riemann sum of the function f tends to a certain value S no matter how it is sampled and divided in the closed interval [a, b], as long as the maximum length of its subinterval is small enough, then the Riemann integral of f in the closed interval [a, b] exists and is defined as the limit S of riemann sum. At this point, the function f is called Riemannian integrable.