Maximal ideal in the ring of continuous functions

Question

Let $R$ be the ring of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ with the usual operations and $I$ the subset of functions $f$ with $f(x_0)=0$ for some $x_0\in\mathbb{R}$.

It's easy to see that $I$ is a prime ideal but I can't see how to show that is maximal and what is the quotient $R/I$.

Answer 1

Hint: One way to show that $I$ is maximal is to show that $R/I$ is a field, so the best way may be to focus on $R/I$. Consider the evaluation map $R \rightarrow \mathbb R$ which sends $f \mapsto f(x_0)$. What is the kernel of this map?