Ideals In $\mathcal{C}[0,1]$

An ideal in $\mathcal{C}[0,1]$ is a subring $I \subset \mathcal{C}[0,1]$ such that for any $f\in I$ and $g \in \mathcal{C}[0,1]$, the product $fg$ is in $I$.

If $I \subset \mathcal{C}[0,1]$ is an ideal such that $I \neq \mathcal{C}[0,1]$, then $I$ is called proper.

A maximal ideal $M \subset \mathcal{C}[0,1]$ is a proper ideal such that whenever $I$ is an ideal with $M \subset I$, in fact $M = I$. $($In other words, $M$ is maximal with respect to the partial order given on ideals of $\mathcal{C}[0,1]$ by set inclusion.$)$

For $x\in [0,1]$, define $M_x := \{f \in \mathcal{C}[0,1] \, | \, f(x) = 0\}.$ Note that $M_x$ is an ideal in $\mathcal{C}[0,1]$.

Answer the following yes-no questions:

• Is $M_x$ is a maximal ideal of $\mathcal{C}[0,1]$ for all $x\in [0,1]$?
• Does there exist a maximal ideal of $\mathcal{C}[0,1]$ that is not of the form $M_x$ for some $x\in [0,1]$?