Find a choice function on $\{\{X,Y\} \vert X,Y \in \mathcal{P}(\mathbb{R})\}$
While reading Adrien Douady's book "Algèbre et Théorie Galoisienne", the first chapter focuses on the axiom of choice and its classic applications, while it's not really the book that focuses on this topic, there is one exercise that stands out.
The question is originally stated as explicitating a choice function on the set $$ E=\{ \{ -g,g \} \vert g\in \mathcal{C} (\mathbb{R})\}$$
In other words, I need to explicite a function $f$ such that $\forall x \in E, f(x)\in x$
Now this is relatively easy without choice thanks to the well-behaved nature of continuity, however then the exercise asks if the same can be done when $g$ ranges over all functions from $\mathbb{R}$ to $\mathbb{R}$, in other words: $$E=\{ \{ -g,g \} \vert g\in \mathbb{R}^{\mathbb{R}})\}$$
Intuitively I want to say it can't be done in $ZF$ at least without the axiom of binary choice, I showed it's equivalent to finding a choice function on the set of all $\{X,Y\}$ with $X,Y \in \mathcal{P}(\mathbb{R})$. This seems like a concrete example of Russell's proverbial "pairs of socks".
To continue on this problem, I was hoping that such a choice function could be use to create a choice function for a more famous problem (like that of $\mathcal{P}(\mathbb{R})$) of which existence is independent from $ZF$ has already been proved.
However maybe forcing provides a direct answer, unfortunately I'm still very new to forcing and don't know if its appropriate in this circumstance.
Any help would be very appreciated.