TITLE: Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?
QUESTION [11 upvotes]: We know the Stone-Weierstrass theorem for locally compact Hausdorff spaces (LCH) which states the following:

Theorem: Suppose $X$ is LCH. A subalgebra $\mathcal{A}$ of $C_0(X)$ is dense if and only if it separates points ($\forall x,y \in X : x\neq y \implies \exists f \in \mathcal{A}: f(x) \neq f(y)$) and vanishes nowhere $(\forall x \in X \exists f \in \mathcal{A} : f(x) \neq 0$) and is closed under complex conjugation1. 

It's also easy to show that $C_b(X)$, the continuous and bounded functions on a topological space $X$, is a Banach space2. It would be obvious to try and show a variant of Stone-Weierstrass for $C_b(X)$ where $X$ is merely Hausdorff, and since it's obviously dense in itself, for such a theorem to exist it would be required that $C_b(X)$ separates points (that it vanishes nowhere is clear: it contains the constant functions). 
I've tried to come up with an example of a Hausdorff space where $C_b(X)$ fails to separate points (this could for example be a space where every non-constant continuous function is unbounded), but to no avail. So does anyone happen to have an instructive example lying around?
PS: It's an interesting exercise to prove that given a LCH space $X$ the continuous functions with compact support, $C_{00}(X)$, separates points and vanishes nowhere. You get to use many theorems from topology in the process of constructing a continuous function such that $f(x)=1$ and $f(y)=0$ given distinct points $x,y \in X$ [Hint: Find a good compact subset and use Urysohn's lemma].
[1]: In the case of real-valued functions this is essentially a no-op, so including it in the theorem statement doesn't hurt.
[2]: [Car00] shows in Lemma 10.8  that $B(X)$, the set of bounded functions on a set $X$ is a Banach space, and it's an immediate corollary from Thm 10.4 that $C_b(X)$ is closed in $B(X)$ for $X$ a metric space. Of course this  generalises readily to the case of $X$ a topological space.

REPLY [2 votes]: I recommend this short 1971 note of T.E. Gantner.  He first proves:
Theorem: Each regular space $Z$ can be embedded as a subspace of a regular space $Q(Z)$ such that every continuous, real-valued function on $Q(Z)$ is constant on $Z$.
He applies the theorem as follows: take $X_0$ to be a one-point space, and inductively, for all $n \in \mathbb{N}$, define $X_{n+1} = Q(X_n)$.  Let $X = \lim_{n \rightarrow \infty} X_n$, endowed with the direct limit (or final) topology.  Then $X$ is an infinite regular space on which every continuous real-valued function is constant.