Euler's broken lines in systems with Carathéodory conditions

We consider Euler's broken lines in a system with its right-hand side measurable in time and investigate their convergence to trajectories of the system. Counterexamples are given that show that partitions with a small diameter do not guarantee the proximity to the funnel of trajectories. For any Carathéodory function, it is suggested to equip the set of closed subsets of the time interval with a metric. We prove that, under conditions close to Carathéodory ones, the convergence with respect to the metric guarantees the convergence of Euler's broken lines to the funnel of solutions of the system. As a consequence, it is shown that if the right-hand side is continuous and the sublinear growth condition is satisfied, then a sufficiently small diameter of the partition guarantees the proximity of Euler's broken line to the funnel of solutions of the system.