I'm looking for an injective function from the set $A$ of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$. Any hints?
I think the other direction is easy: An injective function from $\mathcal{P}(\mathbb{R})$ to $A$ is just a functions that maps all $X \in \mathcal{P}(\mathbb{R})$ to a function $g$ with $g(\mathbb{R})=X$. Is that correct?