The initial-value problem (the Cauchy problem) for an ordinary differential equation of the first order is considered. It is assumed that the right-hand side of the equation is a continuous function defined on a set consisting of a connected open set (a domain) of the two-dimensional Euclidean space, as well as on part of its boundary. It is known that, for any point of the domain, the Peano theorem guarantees the existence of a solution to the Cauchy problem determined on the Peano segment. The sufficient conditions for the existence of a solution to the Cauchy problem set at the boundary point of the domain are formulated, and its existence at some analog of the Peano segment is proved by the Euler polygonal method. Also, the sufficient conditions for the absence of a solution to the Cauchy problem set at the boundary point are presented.