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Let $f:\mathbb{R}\to\mathbb{R}$ is differentiable and $f(0)=0$. Also $\forall x\in \mathbb{R}$ we have $f'(x)=f^2(x)$. Prove that $f(x)=0$, for every $x$.

I tried to use MVT for both derivative and integral. But I got nowhere.

I just found out that

$f$ is increasing. for positive values $f$ is non negative. $\forall x>0$, there exists some $c\in (0,x)$ s.t. $f(x)=xf^2(c).$

Intuitively, it seems one can start by a small interval around zero and show that $f=0$ and so on.

Any comment!