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.
Best Answer
Don't move forward. As often happens, you need to move sideways, backwards, different sideways, forward, sideways, sideways, backwards, sideways, forward in order to actually move forward.
Note that the set of functions from $\Bbb R$ into $\{0,1\}$ is a subset of all real-valued functions, and remember Cantor's theorem.