Sufficient conditions for integrability: the function is continuous or the function is bounded on the interval and has finite discontinuous points. Or function is monotonic in the interval.
Sufficient conditions for the existence of primitive functions: continuity. In addition, the function contains the first discontinuity, so there is no original function, and there is no original function with infinite discontinuity.
Question 1: No, if f(x) has the original function F(x), then F'(x)=f(x). If f(x) is discontinuous at x=c, it is bound to be f(c 0)≠f(c-0), which leads to F'(c 0) ≠. Discontinuous points F'(c 0)=F'(c-0) can be removed, but obviously they are not equal to F'(c)[ for example, F'(c 0)=f(c 0)≠f(c)]. In fact, the function has the discontinuity of the first kind, so there must be no original function.
Question 2: Yes. A finite number of discontinuous points does not affect integrability. If the original function f' (x) = F(x) exists, then according to the nature of the function, the derivative function must be continuous, so F(x) is continuous.
Extended data:
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, if fx) 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.
Let F'(x)=f(x), and f(x) is discontinuous at x=x0, then x0 must be the second kind of discontinuity (for postgraduate mathematics, it can only be the second kind of oscillation discontinuity), not the first kind of discontinuity or the second kind of infinite discontinuity.
When f(x) has the second kind of oscillation discontinuity, it is impossible to determine whether the original function exists, and the conclusion is related to the expression of f(x).
Three conclusions about the existence of primitive function;
If f(x) is continuous, there must be an original function;
If f(x) is discontinuous, there are the first kind of removable and jumping discontinuous points or the second kind of infinite discontinuous points, then there must be no original function in the interval containing the breakpoint here;
If f(x) is discontinuous and there is a second oscillation discontinuity, the original function may or may not exist in the interval containing this breakpoint. ?
Integrability of function product;
Let a function be integrable on an interval, and the product can also be integrable.
Integrability of absolute value of function;
If a function is integrable on an interval, then its absolute value function is integrable, and it satisfies:
References:
Baidu Encyclopedia-Integrable Function
References:
Baidu Encyclopedia-Existence Theorem of Original Function