Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable and $f(0)=0$ s.t. $\forall x \in \mathbb{R}$ we have
$$f'(x)=[f(x)]^2,$$
where $[x]$ is the least integer greater than or equal to $x.$
Show that $f(x)=0$, for every $x\in \mathbb{R}$.
I tried to take the Riemann integral of both sides and I got stuck. I appreciate any help.
By Darboux's Theorem, the derivative of a differentiable function has the intermediate value property. But the derivative of a solution to your equation $f'(x) = \left[ f(x)\right]^2$ only takes integer values. Therefore solutions to this equation must be constant.