Let $A$, $B \in \mathcal{B}(\mathbb{R})$, $y_0 \in B$ be an accumulation point for $B$, and let $f: A \times B \to [-\infty, +\infty]$ be a $\mathcal{B}(A \times B)$-measurable function. Assume that for every $x \in A$, the following limit exists:$$g(x) = \lim_{y \to y_0} f(x,y) \in [-\infty, +\infty].$$
(a) If $B$ is at most countable, can we conclude that $g$ is $\mathcal{B}(A)$-measurable?
(b) Without additional assumptions, can we conclude that $g$ is $\mathcal{B}(A)$-measurable?
For every $C \subseteq \mathbb{R}$, define:$$E_C := \{(x, y) : y = x \in C \}.$$
(c) If $E_C \in \mathcal{B}(\mathbb{R}^2)$, can we deduce that $C \in \mathcal{B}(\mathbb{R})$?
(d) If $C \in \mathcal{B}(\mathbb{R})$, can we deduce that $E_C \in \mathcal{B}(\mathbb{R}^2)$?
Let $h: \mathbb{R}^2 \to \mathbb{R}$ be a function such that, for every $x \in \mathbb{R}$, the function $h(x, \cdot)$ is $\mathcal{B}(\mathbb{R})$-measurable, and for every $y \in \mathbb{R}$, the function $h(\cdot, y)$ is $\mathcal{B}(\mathbb{R})$-measurable.
(e) Can we deduce that $h$ is $\mathcal{B}(\mathbb{R}^2)$-measurable?
I have already solved the points $(c),(d)$ and $(e)$, but I am confused about the first two points, in particular, I don't understand how the hypothesis that $B$ is at most countable affects measurability of $g$ and consequently what difference there is in the question $(b)$. I'm stuck on these two points, any help? :)