Let $S \subset \mathbb{R}^2$ be arbitrary but subject to the following conditions:
$(i)$: $S$ is countably infinite
$(ii): \inf\{||\textbf{x}-\textbf{y}|| : \textbf{x},\textbf{y} \in S \ \text{and} \ \textbf{x} \neq \textbf{y}\} > 0$
$(iii)$: If $\textbf{p} = (x_1, y_1)$ and $\textbf{q} = (x_1, y_2)$ are both elements in $S$, then $y_1 = y_2$.
$(iv)$: The $x$ coordinates are discrete in $\mathbb{R}$. (credit to Daniel Fischer in the comments for this)
Roughly speaking, these are just points in the $xy$ plane which pass the vertical line test and are 'well behaved' in the sense that they're all far apart from each other. It is possible that the second condition is strictly unnecessary for my question.
For any such set, can we construct a $f \in C^{\omega}(\mathbb{R})$ such that $S \subset \{(x, f(x)) : x \in \mathbb{R}\}$? In other words, is there an analytic function which passes through all of the points of $S$?
This is sort of a generalization of the Langrange interpolating polynomials, except with them the set $S$ is strictly finite. In this case, polynomials don't always work. For example, if all but one member of $S$ lies on the line $y=0$, then the function must have an infinite number of zeroes but not be the zero polynomial, hence it can't be a polynomial.