Almost Continuability of Solutions of Differential Equations

We introduce the notion of almost continuability of the solution of the differential equation of first order $dy/dx=f(x,y)$ to the whole real axis. We give a criterion for the almost continuability of solutions for the case in which the right-hand side of the equation is a meromorphic function of one variable $y$: $f(x,y)=g(y)$. As an example, we work out the case of a rational and, in particular, an entire function $g(y)$.