Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{R}$) that have $\mathbb{R}$ as the residue field. There is an obvious bijection from $S$ to the set of functions $\mathbb{R}\to\mathbb{R}$. Is there a locally closed subscheme $Y\subset X$ such that $Y\cap S\subset S$ corresponds exactly to the continuous functions $\mathbb{R}\to\mathbb{R}$?