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.
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?