Let $f: [a,b] \to \mathbb R$. If $f$ is (Riemann-)integrable on $[a,b]$, then define $F: [a,b] \to \mathbb R$ by $$F(x) = \int_a^x f.$$
We have the following:
$$\begin{array}{ccccccc} f \text{ differentiable} & \implies & f \text{ continuous} & \implies & f \text{ integrable} & \implies & ?_1 \\ \Big\Downarrow & & \Big\Downarrow & & \Big\Downarrow & & \Big\Downarrow \\ ?_2 & \implies & F \text{ differentiable} & \implies & F \text{ continuous} & \implies & F \text{ integrable} \end{array}$$
Of course, we can define $$\mathcal F(x) = \int_a^x F, \qquad \mathfrak F(x) = \int_a^x \mathcal F, \qquad \text{etc.}$$ for this chain of implications to grow indefinitely.
My questions are
- ($?_1$) Is there a property of $f$, strictly weaker than integrability, that guarantees that $F$ is integrable?
- ($?_2$) Similar.
Edit: Ignore $?_1$. That was just plain stupidity on my part. If $f$ isn't integrable, then $F$ isn't even defined. Oops.