Corollary 2

If, on I × H, the function f is Lipschitz with respect to a regulated function k(t), and if the right-hand endpoint β of J belongs to I, then u has a left limit at β; if H = E and if f is Lipschitz with respect to a regulated function k(t) on I × E, then J = I.

Indeed, if β ∈ I, there exists a compact neighbourhood V of β in I, such that f(t, x_0) and k(t) are bounded on V; then $\|f(t, x)\| \leq m \|x\| + h$ (m and h constant) on $V \times H$, whence $\|u'(t)\| \leq m \|u(t)\| + h$ on the complement of a countable subset of $V \cap J$, so that $\|u(t)\| \leq m \int_{t_0}^t \|u(s)\|\ ds + q$ ($q$ constant) on $V \cap J$; lemma 2 (IV, p. 170) shows that $\|u(t)\| \leq c\ e^{mt} + d$ ($c$ and $d$ constant) on $V \cap J$, and thus $f(t, u(t))$ remains bounded on $J$, and the corollary results from th. 2 of IV, p. 172.