an example of a non convex ideal

Question

As an example of a non convex ideal we have in Gillman and Jerison, Rings of Continuous Functions, 1976, Exercise 5E(1), the ideal $I= (|\operatorname{id}_{\mathbb R}|)$ in $C(\mathbb R)$. I need to get some details as a verification of this example.

Answer 1

We need to find a function $f(x)$ that is not a multiple of $|x|$ such that $0\leq f(x) \leq | x|$.

One way to do this is to define $f(x)=0$ for $x\leq 0$ and $f(x) = |x|$ for $x\geq 0$. Then, if there is a function $g(x)$ such that $g(x)|x| = f(x)$, we have $g(x) = 0$ for $x<0$ but $g(x)=1$ for $x>0$, so $g$ cannot be continuous.