This is $(7.36)$ exercise of Hewitt & Stromberg - Real and Abstract analysis and i can't figure out the construction.
Let $X$ be any noncompact subset of $\mathbb{R}$. Find a separating family $\mathcal{C}$ in the set of all continuous function on $X$, such that polynomials in the family $\mathcal{C} \cup \{1\}$ are not dense in the set of all continuous function on $X$.
Is this a way to describe the necessary condition about compactness for the set X in the Stone-Weierstrass theorem?
Suppose at first that $X$ is not closed and let $x_0\in \overline{X} \setminus X.$ Define function $\varphi : X\to\mathbb{R} ,$ $\varphi (t) =(t-x_0 )^{-1} $ then $\varphi\in C(X)$ but for any polynomial $W:\mathbb{R}\to\mathbb{R} $ we have $$\left|\left| W-\varphi \right|\right|_{\infty} =\infty.$$ If $X$ is closed then $X$ must be unbounded. So in this case let us define $\psi :X\to\mathbb{R} ,$ $\psi (t) = e^{|t|} $ then $\psi\in C(X)$ but for any polynomial $V:\mathbb{R}\to\mathbb{R} $ we have $$\left|\left| V-\psi \right|\right|_{\infty} =\infty.$$