Regularity of solutions to an abstract inhomogeneous linear differential equation

Let <!-- MATH $T(t),t \geqslant 0$ --> , be a strongly continuous semigroup of linear operators on a Banach space X with infinitesimal generator A satisfying <!-- MATH $T(t)X \subset D(A)$ --> for all 0$">. Let f be a function from <!-- MATH $[0,\infty )$ --> to X of strong bounded variation. It is proved that <!-- MATH $u(t){ = ^{{\text{def}}}}T(t)x + {\smallint ^{t0}}T(t - s)f(s)ds,x \in X$ --> , is strongly differentiable and satisfies <!-- MATH $du(t)/dt = Au(t) + f(t)$ --> for all but a countable number of 0$">.