Euler's broken lines and diameter of partition

We study the conditions on right-hand side of a system that guarantee the convergence of Euler's broken lines to the funnel of solutions of the system for sufficiently small diameter of partition; in particular, the condition that lets us select a subsequence from any sequence of Euler's broken lines that would converge to the solution on a given time interval. We obtain the condition that guarantees the convergence of Euler's broken lines to the funnel of solutions of the system as the diameter of partitions corresponding to the broken lines tends to zero. The condition is specified for a given explicit constant such that for any mapping that is Liepshitz continuous with this constant and maps onto the phase plane, the set of points of discontinuity has the zero Lebesgue measure (on the graph of this mapping). Several examples are given to demonstrate that this condition cannot be relaxed; specifically, there may be no convergence even if, for each trajectory generated by the system, the restriction of the dynamics function to that graph is Riemann integrable; the constant from the condition above can never be decreased either.
\qquad In the paper, Euler's broken lines are embedded into the family of solutions of delay integral equations of the special form, for which, in its own turn, the main result of the paper is proved. It is due to this fact that the results of the paper hold for a broader class of numerical methods, for example, for broken lines with countable number of segments.