I have a pseudometric $\mu$ on $C(\mathbb{I})$ defined by $$\mu(f, g) = |f(x_0) - g(x_0)|.$$ I then take the metric identification of $(M, \mu)$ and am asked what familiar space this metric identification is isometric to?
This question has me pretty much stumped. In my head I am thinking it might be related to the discrete metric, but since I am not sure this could be 100% off-base. Any hints on how to start/what to look at?
Extended HINT: For any $f,g\in C(\Bbb I)$, $\mu(f,g)=0$ if and only if $f(x_0)=g(x_0)$. Thus, if $\sim$ is the corresponding equivalence relation, $f\sim g$ if and only if $f(x_0)=g(x_0)$. Let $[f]$ be the $\sim$-equivalence class of $f$: by definition
$$\begin{align*} [f]&=\{g\in C(\Bbb I):f\sim h\}\\ &=\{g\in C(\Bbb I):f(x_0)=g(x_0)\}\\ &=\{g\in C(\Bbb I):\mu(f,g)=0\}\;. \end{align*}$$
For each $\alpha\in\Bbb R$ let $f_\alpha:\Bbb I\to\Bbb R:x\mapsto\alpha$ be the constant function on $\Bbb I$ with value $\alpha$.
Show that for each $f\in C(\Bbb I)$ there is a unique $\alpha\in\Bbb R$ such that $f\in[f_\alpha]$ and hence such that $[f]=[f_\alpha]$.
If $\alpha,\beta\in\Bbb R$, what is $\rho^*\left([f_\alpha],[f_\beta]\right)$ in terms of $\alpha$ and $\beta$?