The question is as in the title.

Let $H$ be a separable Hilbert space and $f : [0,1] \to H$ be a continuous mapping such that
\begin{equation}
f'(t):=\lim\limits_{\alpha \to 0} \frac{f(t+\alpha)-f(t)}{\alpha} \in H
\end{equation}
exists for almost every $t \in [0,1]$ and $f' : [0,1] \to H$ is Borel-measurable. In the above formula, convergence of the limit is with respect to the norm of $H$.

However, let us further suppose that $H=L^2\bigl([0,1],\mathbb{R}\bigr)$ itself, so that $f(t) : [0,1] \to H$ can be in fact written as $f(t,x) : [0,1] \times [0,1] \to \mathbb{R}$


Then, I am curious about the following:

1. Is it possible for the above derivative $f'(t) : [0,1] \to H$ to be defined almost everywhere on $[0,1] \times [0,1]$ as a real-valued Borel-measurable function?

2. If so, does $f'(t)$ coincide almost everywhere on $[0,1]^2$ with the usual definition of partial derivative of $f(t,x)$ with respect to $t$?


This seems more subtle than I expected and I am quite confused about dealing weith almost everywhere equalities... I posted this question on ME but haven't received any answer yet. all my attempts to figure out myself have failed.. Could anyone please help me?