continuous $\varepsilon$-selection, for every continuous function ${\varepsilon:X\to (0,+\infty)}$.

We proceed as in the proof of Theorem \[theorem-Finite-C-v6:1\], but now using Theorem \[theorem-Near-sel-v9:1\] instead of Theorem \[theorem-Near-sel-v7:1\]. Namely, for a continuous function $\varepsilon:X\to (0,+\infty)$, take open covers $\mathscr{U}_n$, $n<\omega$, of $X$ and mappings $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ as those in Theorem \[theorem-Near-sel-v1:1\] (applied with the embedding relation ${\lhook\joinrel\xrightarrow{\omega}}$ and the function $\varepsilon$). Since $X$ is a paracompact finite $C$-space, by Theorem \[theorem-Selections-IDS-v2:2\], there exists a finite sequence $\mathscr{V}_{n}$, $n\leq k$, of discrete families of nonempty open subsets of $X$ such that each $\mathscr{V}_n$, $n\leq k$, refines $\mathscr{U}_n$, and $\mathscr{V}=\bigcup_{n\leq k}\mathscr{V}_n$ is a cover for $X$. Next, precisely as in the proof of Theorem \[theorem-Finite-C-v6:1\], we define a partial order on $\mathscr{V}$ and mappings $\Phi,\Psi:\mathscr{V}\operatorname{\leadsto}Y$ such that for every $V\in \mathscr{V}$, $$\label{eq:Near-sel-v10:2}
    \begin{cases}
      \Phi(V){\lhook\joinrel\xrightarrow{\omega}}\Psi(V)\subset
      \mathbf{O}_{\varepsilon(p)}(\varphi(p)),\
      \text{whenever  $p\in V$, and }\\
      \Psi(V)\subset \Phi(U),\ \text{whenever $U\in \mathscr{V}$ with
        $U<V$ and $U\cap V\neq {\varnothing}$.}
    \end{cases}$$ Thus, by Theorem \[theorem-Near-sel-v9:1\], there exists a continuous map $h:|\mathscr{N}(\mathscr{V})|\to Y$ satisfying with respect to the oriented simplicial complex $\mathscr{N}(\mathscr{V})$.

We may now finish the proof precisely as before. Namely, since $X$ is paracompact, the cover $\mathscr{V}$ admits a continuous map $g:X\to |\mathscr{N}(\mathscr{V})|$ such that $g(x)\in |\sigma_x|$, $x\in X$, where $\sigma_x=\{V\in \mathscr{V}: x\in 