I am trying to understand why the Fundamental Theorem of Calculus is written the way it is done in this Wikipedia page. So, I go removing hypothesis and check if anything breaks. First check: are there functions $f:[a,b]→ℝ$ and $F:[a,b]→ℝ$ such that $F$ is the antiderivative of $f$ (i.e. $F'(x) = f(x)$ for $x\in(a,b)$) and $$ (1)\quad\quad\quad\quad\quad \int^b_a f(t) dt = F(b) - F(a) $$ yet $f$ is not Riemann integrable? The left side of equation (1) would not even have meaning without the hypothesis ‘$f$ is Riemann integrable’. Trying to savage the maimed theorem, let us say that the integral can be Lebesgue. Then, the usual ‘$f(x) = 0$ when $x$ is irrational and $f(x) = 1$ when $x$ is rational’ comes to mind. But that function does not have a derivative. Hence two questions:
Are the derivative and the Riemann integral so related to each other that the FTC cannot be expanded to other types of integrals?
If not, what is the anti-Lebesgue-integral? (in the same way that derivation would be the anti-Riemann-integral)
