Prove that the ring $C^0[a,b]$ is not Euclidean

Question

I'm trying to prove that the ring $C^0[a,b] = R$ is not Euclidean. I think I need to use proof by contradiction for that. Suppode there is a Euclidean function $d$ on $R$. Then $\forall a,b \in R, b \neq {0}$ there exist $q, r \in R$ such that $a = qb+r$ so that $r = 0$ or $d(r)<d(b)$

And pretty much here I'm stuck. I know that the quotient of two continuous functions is also a continuous function, except for those values of x for which the denominator vanishes, so I'm thinking I need to use this somehow, but I am really unsure of how to proceed and would like some help.

Answer 1

The ring $\mathcal{C}^0[a,b]$ is not an integral domain. That is why it is not Euclidean.