Consider $f$ from $\mathbb{R}$ to $\mathbb{R}$.
Suppose that for each $r\in \mathbb{R}$ I collect $f(r)$ in $\mathcal{A}_f$ (if for any $r\in \mathbb{R}, r'\in \mathbb{R}$ we have $f(r)=f(r')$, I put both $f(r), f(r')$ in $\mathcal{A}_f$).
Which mathematical object is $\mathcal{A}_f$? Uncountable infinite sequence (does it make any sense)? An interval of $\mathbb{R}$ (but then there are some points which could have more weight because they appear more than once)?
EDIT: I am updating my question thanks to the illuminating comments below which suggest that $\mathcal{A}_f$ is an uncountably infinite (multi)set.
I want to go deeper and ask
Is there any specific name relating $\mathcal{A}_f$ to $f$ (e.g., the multiset generated by the function $f$)
Let $\mathcal{F}$ be the collection of all functions from $\mathbb{R}$ to $\mathbb{R}$. Let $\mathcal{A}$ be the collection of $\mathcal{A}_f$ $\forall f \in \mathcal{F}$, where $\mathcal{A}_f$ is defined above. Does the object $\mathcal{A}$ make sense?
Suppose I write the object $\mathcal{A}\times \{0,1\}$ where $\times$ denotes the Cartesian product. Is this Cartesian product well defined in math? Intuitively I just want to combine $\{0,1\}$ with any multiset in $\mathcal{A}$. Also, can I (sloppily) call an element of $\mathcal{A}\times \{0,1\}$ "a point", or is there any other specific name I should use?