Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

      Hence,

.

      The next result, borrowed from [7, Lemma 2.3], specifies the supremum of a function

.

      

      Proposition 1.6: Let . Then where either , for some , or , for some , or for some .

      Proof. Let

. Since

,

. By the definition of the supremum, for all

, one can choose

such that

which implies

Since

, there exists a subsequence

such that

as

. Since

is regulated,

belongs to and the proof is complete.

      The composition of regulated functions may not be a regulated function as shown by the next example proposed by Dieudonné as an exercise. See, for instance, [58, Problem 2, p. 140].

      

      Example 1.7: Consider, for instance, functions given by , for , and , that is, is the sign function. Both and belong to . However, the composition does not.

      The next result, borrowed from [209, Theorem 10.11], gives us an interesting property of left‐continuous regulated functions. Such result will be used in Chapters 8 and 11. We state it here without any proof.

      

      Proposition 1.8: Let . If for every , there exists such that for every , we have , then

      We end this first section by introducing some notation for certain spaces of regulated functions defined on unbounded intervals of the real line . Given , we denote by the space of regulated functions from to . In order to obtain a Banach space, we can intersect the space