  This paper is inspired by the paper of Leonetti, Russo and Somaglia
[\textit{Dense lineability and spaceability in certain subsets of
$\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the
lineability problems raised therein. It concerns the properties of
$\ell_\infty$ subsets defined by cluster points of sequences. Using the fact
that the set of cluster points of a sequence $x$ depends only on its
equivalence class in $\ell_\infty/c_0$ and that the quotient space
$\ell_\infty/c_0$ is isometrically isomorphic to
$C(\beta\mathbb{N}\setminus\mathbb{N})$, we are able to translate lineability
problems from $\ell_\infty$ to $C(\beta\mathbb{N}\setminus\mathbb{N})$. We
prove that for a compact space $K$ with properties similar to those of
$\beta\mathbb{N}\setminus\mathbb{N}$, the sets of continuous functions $f$ in
$C(K)$ with $\vert\operatorname{rng}(f)\vert=\omega$ and those $f$ with
$\vert\operatorname{rng}(f)\vert=\mathfrak c$ contain, up to zero function, an
isometric copy of $c_0(\kappa)$ for uncountable cardinal $\kappa$. Specializing
those results to some closed subspaces $K$ of
$\beta\mathbb{N}\setminus\mathbb{N}$ we are able to generalize known results to
their ideal versions.
