Roadmap through the three strengths of FTC I and the five Riemann strengths of FTC II

There are three progressively broader Riemann forms of the first fundamental theorem. The continuous-integrand form says that the integral function is differentiable everywhere. The pointwise form The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive needs continuity only at the point where the derivative is taken. The almost-everywhere consequence on this page combines that pointwise theorem with the Riemann integrability criterion; the later Lebesgue theory gives the corresponding absolutely-continuous formulation.

For Newton--Leibniz, the continuously differentiable working form is contained in The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, whose published statement already allows an arbitrary integrable derivative on the closed interval. This page successively removes endpoint differentiability, permits finitely many exceptional interior points, and then permits a countable exceptional set by Botsko's theorem. The later absolutely-continuous theorem is stronger in a different direction and is not used here.