# Bochner integrability of mappings of Bochner integrable functions

Suppose I have a Bochner integrable function, $$t\mapsto u(t)\in X$$, with $$X$$ a separable Banach space, and $$0\leq t\leq T<\infty$$. If I introduce a mapping $$f:[0,T]\times X\to X$$, under what assumptions will $$t\mapsto f(t,u(t))$$ be Bochner integrable? Is joint measurabiity of $$f$$ sufficient? Continuity?