  Let g denote a real analytic function on an open subset U of Euclidean space,
and let S denote the boundary points of U where g does not admit a local
analytic extension. We show that if g is semialgebraic (respectively, globally
subanalytic), then S is semialgebraic (respectively, subanalytic) and g extends
to a neighbourhood of cl(U)\S as an analytic function that is semialgebraic
(respectively, globally subanalytic). (In the general subanalytic case, S is
not necessarily subanalytic.) Our proof depends on controlling the radii of
convergence of power series G centred at points in the image of an analytic
mapping, in terms of the radii of convergence of the pull-backs of G at points
of the source.
