I was discussing with friends the astounding fact that $\mathbb R$ and the set of real continuous functions were equipotent.
I asked for a proof that $\mathbb R$ and $\mathbb R ^{\mathbb R}$ are not equipotent. Someone asserted that given any set $A \subset \mathbb R$, there exists a function $f\in \mathbb R ^{\mathbb R}$ such that the codomain of $f$ is exactly $A$.
Is that true ?
Here's something that comes immediately to mind:
Give some set $A\subset \mathbb{R}$, pick any element $a\in A$ and define the function $f\in \mathbb{R}^{\mathbb{R}}$ by:
$$ f(x) = \left\{ \begin{array}{lr} x & : x \in A\\ a & : x \notin A \end{array} \right. $$