Given a sequence of continuous functions ${{f_n}}$, define the **varicontinuity index** $$V({f_n}): \mathbb{R} \to [0, \infty]$$ by
\begin{split}
V({f_n})(x) &=\sup \Big\{\varepsilon > 0\big|\; \forall d > 0, \exists y, z \in B_d (x),\; \exists n\in \mathbb{N}: |f_n (y) - f_n (z)|\geq \varepsilon\Big\}\\
&=\sup \left\{\varepsilon > 0\big|\; \forall d > 0\sup_{\substack{y,z \in B_d(x),\\ n\in\mathbb{N}}} |f_n(y)-f_n(z)|\geq \varepsilon\right\}\\
&=\inf_{d>0}\sup_{\substack{y,z \in B_d(x),\\ n\in\mathbb{N}}} |f_n(y)-f_n(z)|\\
& =\inf_{d>0}\sup_{n\in\mathbb{N}}\mathrm{osc}\big(\,f_n,\;B_d(x)\big)
\end{split}
where $\mathrm{osc}(f,S)$ denotes the oscillation of $f$ on a set $S$.

I want to investigate $V$ itself as an operator $$V: C(\mathbb{R})^\mathbb{N} \to [0, \infty]^\mathbb{R}$$

What are the relevant topologies/structures to put on the range and codomain? What kind of spaces are they under these structures? And in general what tools from functional analysis are relevant to investigating $V$?