Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov’s introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if f:[a,bλ]→Rn is of bounded variation then f is the sum of an absolute continuous function fa and a singular function fs where the total variation of fs generates a singular measure τ and fs is absolute continuous with respect to τ. In this paper we prove that a function of bounded variation f has two representations: one is f which was described with an absolute continuous part with respect to the Lebesgue measure λ, while in the other an integral with respect to τ forms the absolute continuous part and t(τ) defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain [a,bν].

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Due to noncontinuous solution, impulsive differential equations with delay may have a measurable right side and not a continuous one. In order to support handling impulsive differential equations with delay like in other chapters of differential equations, we formulated and proved existence and uniqueness theorems for impulsive differential equations with measurable right sides following Caratheodory’s techniques. The new setup had an impact on the formulation of initial value problems (IVP), the continuation of solutions, and the structure of the system of trajectories. (a) We have two impulsive differential equations to solve with one IVP (φσ0=ξ0) which selects one of the impulsive differential equations by the position of σ0 in a,bν. Solving the selected IVP fully determines the solution on the other scale with a possible delay. (b) The solutions can be continued at each point of α,β×Ω0≕Ω by the conditions in the existence theorem. (c) These changes alter the flow of solutions into a directed tree. This tree however is an in-tree which offers a modelling tool to study among other interactions of generations.