In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question:
In terms of functions, what does a quotient ring mean?
When we have a quotient ring of a polynomial ring: $$\Bbb R[x]/\langle f \rangle$$
We are setting $f\equiv 0\pmod f$
But when we have for example the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. How do I think of a quotient ring of this?