Function integrable:

The sufficient condition of integrability is 1, and the function is continuous in the closed interval; 2. The function is bounded on the closed interval and has only a finite number of discontinuous points; 3. The function is monotone in the closed interval; It can be seen that these three conditions are parallel, and any one is a sufficient condition for the function to be integrable.

Original function exists:

The existence theorem of primitive function is that if f(x) is continuous on [a, b], primitive function must exist. This condition is a sufficient condition, not a necessary condition. That is to say, if f(x) has an original function, it cannot be deduced that f(x) is continuous on [a, b]. Because the elementary function is continuous in the defined interval, the elementary school has the original function in the defined interval. It should be noted that the derivative of an elementary function must be an elementary function, and the original function of an elementary function is not necessarily an elementary function.