Honestly this is just a case of me not being great at constructing 'pathological' differentiable functions and I am not sure how to search for what I am looking for. This is related to an algebraic method of constructing cotangents that I have seen before, but it has been a long time so I don't remember the details.
Let $D$ be the set of functions $f$ on $\Bbb{R}$ which are differentiable at $0$ and have $f(0)=0$.
Suppose $f_0\in D$ has $f_0'(0)=0$.
Let $D\cdot D$ be the set of finite sums of products of elements of $D$. (i.e. the product ideal)
Is it the case that $f_0\in D\cdot D$? If not is there a counterexample?
It is easy to show that $f_0\in C\cdot D$ where $C$ is the set of continuous functions $f$ with $f(0)=0$ but I can't see if it is true in the restricted case.
Any help or references would be appreciated