  Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the
metric $g$ is real-analytic, it is shown that if the geodesic ray transform of
a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics,
then $f=0$ on the set of points lying on these geodesics. The approach is based
on a microlocal version of unique continuation of analytic functions.
