Pavel Pyrih

Finely holomorphic functions and quasi-analytic classes

AbstractWe study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf:ℝ → ℂ for which there exists a quasi-analytic classC{Mn} containingf. Let ℱ denote the set of complex functionsf:ℝ → ℂ for which there exist a fine domainU containing the real line ℝ and a function

Half-homogeneous indecomposable circle-like continuum

\tilde f

Non-cut, shore and non-block points in continua

finely holomorphic onU satisfyingf(x)=

ON THE POWER PROPERTY OF THE DENSITY TOPOLOGY IN THE PLANE

\tilde f

A lambda-dendroid with two shore points whose union is not a shore set

(x) for allx ∈ ℝ. The “power” of unique continuation is incomparable in these two cases (I\ℱ is non-empty, ℱ\I is non-empty).

On blockers in continua

We consider fine topology in the complex plane C and finely harmonic morphisms. We use oriented Jordan curves in the plane to prove that for a finely locally injective finely harmonic morphism f in a fine domain in C, either f or f is a finely holomorphic function. This partially extends result by Fuglede, who considered a kind of continuity for the fine derivatives of the finely harmonic morphism. As a consequence of this we obtain a both necessary and sufficient condition for a function f to be finely holomorphic or finely antiholomorphic. We do not know if the condition of finely local injectivity (q.e.) is automatically fulfilled by any non-constant finely harmonic morphism.