  We discuss the quasianalytic properties of various spaces of functions
suitable for one-dimensional small divisor problems. These spaces are formed of
functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in
the Whitney sense), as is the solution of a linear or non-linear small divisor
problem when viewed as a function of the multiplier (the intersection of K_j
with the unit circle is defined by a Diophantine-type condition, so as to avoid
the divergence caused by roots of unity). It turns out that a kind of
generalized analytic continuation through the unit circle is possible under
suitable conditions on the K_j's.
