Let $f\in C([0,1],[0,1])$, such that: $$\forall x\in [0,1],\exists k\in \mathbb N, f^{\circ k}(x)=0.$$

Is it true that $f$ nilpotent ?

PS : $f^{\circ 2}(x)=f\circ f (x)$

Let $f\in C([0,1],[0,1])$ be such that: $$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$

Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)?

Here $f^{\circ k}$ denotes the $k$th iterate of $f$.

Let $f\in C([0,1],[0,1])$, such that: $$\forall x\in [0,1],\exists k\in \mathbb N, f^k(x)=0.$$

Is it true that $f$ nilpotent ?

PS : $f^2(x)=f\circ f (x)$

Let $f\in C([0,1],[0,1])$, such that: $$\forall x\in [0,1],\exists k\in \mathbb N, f^{\circ k}(x)=0.$$

Is it true that $f$ nilpotent ?

PS : $f^{\circ 2}(x)=f\circ f (x)$