A necessary condition to $F'(x)=f(x)$ for a continuous function $f$ 
Theorem:  Consider ,
$$F(x)=\int_a^xf(t)\,dt$$
If the function $f:[a,b]\to \mathbb R$ is continuous then , $F(x)$ is differentiable and $F'(x)=f(x).$

I know that the continuity condition of $f$ is sufficient condition.
That means there exists a discontinuous function $f$ for which this $F'(x)=f(x)$.
My Question:
Does there exist a necessary condition for this ?
$$OR$$
After imposing which extra condition on $f$  it is necessary that $F'(x)=f(x)$ ?

Let $f$ be Riemann integrable on $[a, b]$. Then the function $$F(x) = \int_{a}^{x}f(t)\,dt$$ is known to be continuous and of bounded variation on $[a, b]$. Apart from these properties of $F$ it is possible to ensure differentiability of $F$ at points where $f$ is continuous.
Another point to note is that since $F(x)$ is of bounded variation it possesses a derivative almost everywhere i.e. $F$ is differentiable on $[a, b] - E_{1}$ where $E_{1}$ is a subset of $[a, b]$ of measure zero. Similarly since $f$ is Riemann integrable on $[a, b]$, it follows that $f$ is continuous almost everywhere on $[a, b]$ so that there is a subset $E_{2}$ of $[a, b]$ of measure zero such that $f$ is continuous on $[a, b] - E_{2}$.
Since $E = E_{1} \cup E_{2}$ is also of measure zero, it follows that $F'(x) = f(x)$ for all $x \in [a, b] - E$. Thus the relation $F'(x) = f(x)$ holds almost everywhere in $[a, b]$.
Also note that $E_{1} \subseteq E_{2}$. I suspect there are examples where $E_{1} \subset E_{2}$ but I don't have an example right now with me.
