We need to show that the set of real numbers and the set of Real valued functions whose domain is $\mathbb R$ are not similar (equinumerous).
Let $\mathbb R$ denote the set of real numbers and $S$ denote the set of real valued functions whose domain is $\mathbb R$.
Suppose $\mathbb R$ and $S$ are equinumerous. Then, $~\exists ~$ a one-one onto function $f :\mathbb R \rightarrow S $ such that $f( \mathbb R) = S$.
Let $a \in \mathbb R$, then let $g_a$ be the associated real valued function with $a$. Thus :
$f(a) = g_a$.
To bring a contradiction, I think we should show that $f$ is either not one-one or onto.
How do I move forward?
Thank you for your help.