Characterize the function that can be expressed as a Riemann integral of a certain function

Question

We know that Lebesgue integral theory gave a very useful characterization of the dynamic between integration and differentiation, and the culmination is

If $F$ is absolutely continuous on $[a,b]$, then $F'$ exists almost everywhere and is [Lebesgue] integrable. Moreover, for all $x \in [a,b]$, $$ F(x) - F(a) = \int_a^x F'(t)\ \mathrm dt. $$

Then what about analogs [if exists] in Riemann integral theory ? We know that $F$ is Riemann integrable on $[a,b]$ iff it is continuous almost everywhere and bounded. Then do we have any equivalent condition such that there is some $f$ Riemann integrable on $[a,b]$ and $$ \int_a^x f(t)\ \mathrm dt = F(x) - F(a) \quad [x \in [a,b]]? $$

For example, can we prove or disprove the following assertion

$F$ is continuous and a.e. differentiable on a compact interval $[a,b]$ with $F'$ be bounded and a.e. continuous iff there is some Riemann integrable function $f$ such that $$ \int_a^x f(t)\ \mathrm dt = F(x) - F(a) \quad [x \in [a,b]]? $$

All discussions are welcome. Thanks in advance.

UPDATE

Thanks for the discussion so far. Now that the "example" is disproved, could we find any other nontrivial sufficient conditions that could make a function be a Riemann integral function of some certain function?

Answer 1

Consider the Cantor "devil's staircase" function. This is continuous, and differentiable a.e., on $[0,1]$, and where that derivative exists it is bounded and continuous (in fact $0$). But $F$ is not the integral of that derivative.