A function $f:\mathbb{R}\to\mathbb{R}$ such that $|f(x)|\leq x^2$ for all $x$ is differentiable at $0$.
There are no differentiable functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)=0$ and $x=f(x)g(x)$ for all $x$.
If $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq(x-y)^2$ for all $x$ and $y$, then $f$ is constant.