  We study the convergence of a formal power series of two variables if its
restrictions on curves belonging to a certain family are convergent. Also
analyticity of a given $C^\infty $ function $f$ is proved when the restriction
of $f$ on analytic curves belonging to some family is analytic. Our results
generalize two known statements: a theorem of P. Lelong and the Bochnak-Siciak
Theorem. The questions we study fall into the category of "Osgood-Hartogs-type"
problems.
