What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$? What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$?
I want to say that it is at least $2^{\aleph_0}$, but I have no idea how to sharply pin it down otherwise.

HINT: The cardinality of $\Bbb{R^R}$ (all the functions from $\Bbb R$ to itself, not just the continuous ones or whatnot) is $2^{2^{\aleph_0}}$. And note that if $V$ is a vector space over an infinite field, with basis $B$, then $|V|=\max\{|B|,|F|\}$.
