Hello!

Can anyone help me solve the problems? It paralyzes my work

It is obvious in the first two problems that $\displaystyle f'(0)=0$ and $\displaystyle f$ is Lipschitz. But I can't go any further.

1) A continuously differentiable function $\displaystyle f$ is defined on $\displaystyle \mathbb{R}$ such that $\displaystyle f(0) = 0$ and $\displaystyle |f'(x)| \leq |f(x)|$ for all $\displaystyle x\in \mathbb{R}$. Show that $\displaystyle f$ is constant.

2) Suppose that for some positive constant $\displaystyle M$, $\displaystyle |f'(x)|\leq M|f(x)|$, $\displaystyle x\in[0,1]$. Show that if $\displaystyle f(0)=0$, then $\displaystyle f(x) =0$ for any $\displaystyle x$, $\displaystyle 0\leq x\leq 1$.

3) Show that there exists no real-valued function $\displaystyle f$ such that $\displaystyle f(x)>0$ and $\displaystyle f'(x)=f(f(x))$ for all $\displaystyle x$.

Thank you very much!