Fix a Banach space $X$, an open subset $U \subset X$, and a measure space $(\Omega, \mathcal A, \mu)$. Let $f: U \times \Omega \to \mathcal R$ be continuously Frechet-differentiable in the first argument and let it and its Frechet derivative be measurable in the second argument.
Now fix $u \in U$, and suppose that $f(u,\omega)$ and, for all $x \in U$, $Df(u,\omega)(x)$ are Lebesgue-integrable over $\omega$. Does it follow that $f(\tilde u,\omega)$ is integrable for all $\tilde u$ in some neighborhood of $u$?
I suspect not but I cannot construct a counterexample, and this would be a very useful fact for me.
Nevermind! Here is a counter-example:
$X = \mathbb R$, $\Omega = (0,1]$ with Lebesgue measure.
$$f(x,\omega) = x^2 \frac{1}{\omega}$$