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 $V$?

  • $\begingroup$ 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
  • 1
    $\begingroup$ Thanks so much for the edit, Liviu! $\endgroup$ – James Baxter Dec 27 '18 at 15:10

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.