Theorem(The Fundamental Theorem of Calculus, TFC)

Letfbe a function on[ Assume thata,b].fis Riemann integrable on[ and that it also has an antiderivativea,b]Fon[ Thena,b].

In short, this says that if a function has both the Riemann and Newton integral, they must be connected by the Newton-Leibniz formula. Note that the Riemann and Newton integrability in this general version is less restrictive than the continuity we required in the main text, so this version is more general and therefore more powerful. However, in actual problems we usually use continuity anyway because it is the easiest way out and we usually get continuous functions anyway.

Similarly, the first version of TFC we stated in the main text is true in more generality:

Theorem(The Fundamental Theorem of Calculus I, TFC 1)

Letfbe a Riemann integrable function on[ leta,b],cbelong to[ Fora,b].xfrom[ we definea,b]Then

Fis continuous on( it has derivative from the right at all points froma,b),[ and derivative from the left at all points froma,b)( Moreover,a,b].and for every

xfrom( we havea,b)

Recall that for instance
*F* ′_{+}(*a*)*F* at *a*, and
*f* (*a*_{+})*f* at
*a* from the right. So the first statement in fact means that