AUTOMATIC CONTINUITY VIA ANALYTIC THINNING

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.