  We show that a certain simply-stated notion of "analytic completeness" of the
image of a real analytic map implies the map admits no analytic extension. We
also give a useful criterion for that notion of analytic completeness by
defining arc-properness of continuous maps, which can be considered as a very
weak version of properness. As an application, we judge the analytic
completeness of a certain class of constant mean curvature surfaces (the
so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space.
