I was doing a homework assignment, the problem is the next:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a $C^{1}$ function such that there is no $x\in{\mathbb{R}}$ with $f(x)=f'(x)=0$. Show that the set $$S=\lbrace{ x \in{[0,1]}: f(x)=0} \rbrace $$ is finite.
I already prove it, but i was trying to find an example of a function $f$ that is no $C^{1}$, I mean at least differentiable in $[0,1]$ but not continuously differentiable and such that $f(x)=f'(x)=0$ but S is not a finite set. Any ideas?