Question:
Let $(X, d)$ be a compact metric space. For a given $x_0 \in X$, define $C_{x_0}(X,\mathbb{R})$ by $$C_{x_0}(X,\mathbb{R}) = \{f \in C(X, d):f(x_0) = 0\}$$
Note that $C(X,\mathbb{R})$ denotes all bounded and continuous functions from $X \rightarrow \mathbb{R}$ and the topology is supnorm
Show that:
(1) $C_{x_0}(X,\mathbb{R})$ is closed in $C(X,\mathbb{R})$
(2) $C_{x_0}(X,\mathbb{R})$ is an ideal of $C(X,\mathbb{R})$
Thought:
For first problem, consider a linear function $l(f) = f(x_0)$, then $l$ is continuous
Also, we know that $C_{x_0} = l^{-1}(\{0\})$, which is the preimage of a closed set under a continuous mapping
Therefore, $C_{x_0}$ is closed in $C(X, \mathbb{R})$
Intuitively, I understand why $l$ is continuous, but how to prove it rigorously? For question (2), I know that $C_{x_0}(X,\mathbb{R})$ is actually a subspace of $C(X,\mathbb{R})$, which is a commutative algebra with a multiplication identity. Then what should I do with this statement?