What are the conditions in the definition of Caratheodory functions?
Let $f\colon T\times X\to \mathbb{R}$, where $T\subset \mathbb{R}$ and $X$ is real Banach space.
1) $f(\cdot,x)$ is measurable for every $x\in X$,
2) $f(t,\cdot)$ is continuous for almost every $t\in T$,
3) $f(\cdot,x)\in L(T)$.
Are there any more?