= Cardinality of continuous real functions
{title2=$|C(\mathbb R,\mathbb R)|=2^{\aleph_0}$}
The <set> of <continuous functions> $\mathbb R\to\mathbb R$ has <cardinality> $2^{\aleph_0}$. <Constant functions> provide the lower bound. Restriction to the <dense subset>[dense set] $\mathbb Q$ is <injection>[injective], since <continuity> determines a <function> from its values there. <sequence>[Real sequences] have <cardinality> $2^{\aleph_0}$, by encoding a <sequence> using its countably many rational lower cuts. Restriction therefore supplies the <upper and lower bounds>[upper bound] and <Cantor-Schröder-Bernstein theorem>[Cantor-Schröder-Bernstein] gives <logical equality>[equality].
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