I was just wondering the number of all explicit functions , whatever it is continuous or discontinuous, or it has a graph or not.
As far as I know, it seems that $\mathbb{R}^{\mathbb{R}}$ which denotes all functions from reals from reals has a cardinality of $\aleph_2 $, with my attempt like $$card(\mathbb{R}^{\mathbb{R}})=\aleph_1 ^{\aleph_1 } =2^{\aleph_0 \cdot 2^{\aleph_0 }}<2^{2^{2^{\aleph }}}=\aleph_3 $$ If we accepted GCH (the Generalized Continuum Hypothesis) the result followed trivially. But when it comes to "all" functions with no description of domain, I have literally no idea.
Could anyone please help me? Thanks in advance.