Lineability of functions in $C(K)$ with specified range

This paper is inspired by the paper of Leonetti, Russo and Somaglia [$\mathit{\text{Dense}}\mathit{\text{lineability}}\mathit{\text{and}}\mathit{\text{spaceability}}\mathit{\text{in}}\mathit{\text{certain}}\mathit{\text{subsets}}\mathit{\text{of}}{\mathrm{\ell }}_{\mathrm{\infty }}\mathit{\text{.}}$ Bull. London Math. Soc., 55: 2283–2303 (2023)] and the lineability problems raised therein. It concerns the properties of ${\mathrm{\ell }}_{\mathrm{\infty }}$ 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 ${\mathrm{\ell }}_{\mathrm{\infty }}\mathrm{/}{c}_0$ and that the quotient space ${\mathrm{\ell }}_{\mathrm{\infty }}\mathrm{/}{c}_0$ is isometrically isomorphic to $C(\beta \mathbb{N}\setminus \mathbb{N})$, we are able to translate lineability problems from ${\mathrm{\ell }}_{\mathrm{\infty }}$ 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 $\mathrm{\mid }\mathrm{rng}(f)\mathrm{\mid }=\omega$ and those $f$ with $\mathrm{\mid }\mathrm{rng}(f)\mathrm{\mid }=\mathfrak{c}$ 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.