For what condition on $f$ is $F(x)= \int_0^\infty \frac{\arctan(xf(t))}{t^2+1}dt$ differentiable?

Question

Let $f:\Bbb R_+\to \Bbb R_+$ be a borel measurable function. Then for $x\ge 0$ let

$$F(x)= \int_0^\infty \frac{\arctan(xf(t))}{t^2+1}dt$$

1. Prove that $F$ is continuous on $[0,\infty)$ and compute $\lim\limits_{x\to\infty}F(x).$
2. Prove the differentiability of $F$ on $(0,\infty) $
3. What is necessary and sufficient condition on $f$ under which one has the differentiability of $F$ at $x=0?$

Thanks to the Lebesgue dominated convergence theorem I was able to manage the twofirst questions and I proved that $$\lim\limits_{x\to\infty}\int_0^\infty \frac{\arctan(xf(t))}{t^2+1}dt=\frac{\pi}{2}$$

Can some help with the last question ?

3. What is necessary and sufficient condition on $f$ under which one has the differentiability of $F$ at $x=0?$