Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2007-24

October 2007 (Revised January 2009)

Automatic continuity via analytic thinning

N. H. Bingham and A. J. Ostaszewski

We use Choquet's analytic capacitability theorem and the Kestelman-Borwein-Ditor theorem (on the inclusion of null sequences by translation) to derive results on `analytic automaticity' -- for instance, a stronger common generalization of the Jones/Kominek theorems that an additive function whose restriction is continuous/bounded on an analytic set T spanning R (e.g., containing a Hamel basis) is continuous on R. We obtain results on `compact spannability' -- the ability of compact sets to span R. From this, we derive Jones' Theorem from Kominek's. We cite several applications including the Uniform Convergence Theorem of regular variation.

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