Ideal in the ring of continuous functions [0,1] to $\mathbb{R}$

Question

I know the following question is easy, but I don't understand something

Let $R=C\left(\left[0,1\right]\right)\to \mathbb{R}$ be the ring of continuous functions. for all $z \in [0,1]: M_{z}=\left\{ f\in C\left(\left[0,1\right]\right)\mid f\left(z\right)=0\right\}$ Show that $M_z$ is an ideal.

I tried to prove that $M_z$ is closed under multipication of elements in $R$.

Let $f\in M_z$ and $g\in R$:

$(g\circ f)(z)=g(f(z))=g(0)=^? 0$

And I'm not sure how to proceed. What am I doing wrong?

Thank you.