This topic should be viewed as F(x)=f'(x), so the condition of the topic becomes that the limits of F(x) at the left and right ends of x=x0 exist and are equal, but if F(x) wants to be continuous, there must be a condition that the left and right limits must be equal to F(x0), but the topic is not mentioned, which means that F(x) is not necessarily.

And f(x) is the derivative of F(x). In either case, we can get f(x) by integrating F(x), that is, we can find the area. We know that the derivative function is still integrable for a finite number of discontinuous points, so f(x) is defined as x=x0, but it does not necessarily have a derivative. In addition, from the point of view of calculating the area by integrating F(x),