Help with homomorphisms on the ring of continuous functions

Question

Let $C$ be the ring of continuous functions $f:\Bbb R \to \Bbb R$ with addition and multiplication defined pointwise. Let $J=\{f \in C:f(s)=0\}$, where $s$ is some fixed integer. Then $J$ is an ideal. I want to show that $C/J$ is isomorphic to some well known ring. I know the First Isomorphism Theorem should be used.

I am having trouble in even defining a homomorphism from $C$ to some other ring, never mind finding a homomorphism for which $J$ is the kernel. Any help would be much appreciated.

Thanks.

Answer 1

Edited:

You want $\phi: C \to R$ so that $\phi(f)=0$ if and only if $f(s)=0$.

Isn't it obvious what $\phi(f)$ should be?

Added Once you realize that $\phi(f)$ should be $f(s)$, then your $R$ must contain all real numbers. Moreover, you want $\phi$ to be onto, thus $R$ must contain all real numbers and nothing more...$R= \mathbb R$...

So, to sum it up

$$\phi: C \to \mathbb R \,;\, \phi(f)=f(s)$$

is the function you need. Now check that this works....