Let $X$ be a set of cardinality $\kappa$ such that $\kappa^\omega=\kappa$. We
prove that the linear algebra $\mathbb{R}^X$ (or $\mathbb{C}^X$) contains a
free linear algebra with $2^\kappa$ generators. Using this, we prove several
algebrability results for spaces $\mathbb{C}^\mathbb{C}$ and
$\mathbb{R}^\mathbb{R}$. In particular, we show that the set of all perfectly
everywhere surjective functions $f:\mathbb{C}\to\mathbb{C}$ is strongly
$2^\mathfrak{c}$-algebrable. We also show that the set of all functions
$f:\mathbb{R}\to\mathbb{R}$ whose sets of continuity points equals some fixed
$G_\delta$ set $G$ is strongly $2^\mathfrak{c}$-algebrable if and only if
$\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself
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