Which is the cardinality of the set of all functions $\mathbb{R} \to \mathbb{R}$?

Question

Which is the cardinality of the set of all functions $\mathbb{R} \to \mathbb{R}$ ?

For the relations in $\mathbb{R}$ I think the cardinality is the cardinality of $\mathcal{P(\mathbb{R^2})}$ , but I'm not sure.

Answer 1

$|\mathbb{R}^{\mathbb{R}}|=(2^{\aleph_0})^{(2^{\aleph_0})}=2^{\aleph_02^{\aleph_0}}=2^{2^{\aleph_0}}=^{*}2^{\aleph_1}=\aleph_2$, the last equalities is using the generalized continuum hypothesis.

Answer 2

The cardinality of $\mathbb{R}$ is $c=2^{\aleph_0}$ (continuum). BY DEFINITION, $c^c$ is precisely what you ask for. One can show that $c^c=2^c$ (which is $\aleph_2$ if you care to assume the generalized continuum hypothesis)