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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 $V$?

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  • $\begingroup$ How do I include curly braces (e.g. for set builder notation or sequences) without it being interpreted as part of the code? With that I’ll be able to fix the set builder part... $\endgroup$ – James Baxter Dec 27 '18 at 13:59
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    $\begingroup$ Thanks so much for the edit, Liviu! $\endgroup$ – James Baxter Dec 27 '18 at 15:10

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