= Solution
Write $C(\mathbb R,\mathbb R)$ for the <set> of <continuous real-valued functions> on the <real line>. <Constant functions> give an <injection> of $\mathbb R$ into this <set>. In the other direction, restriction to the <rational numbers> is <injective>. Indeed, if <continuous functions> $f,g$ agree at every rational and $x$ is real, choose <rational numbers> $q_n\to x$. <Continuity> gives
$$
f(x)=\lim_n f(q_n)=\lim_n g(q_n)=g(x).
$$
Thus restriction embeds $C(\mathbb R,\mathbb R)$ into $\mathbb R^{\mathbb Q}$, which has the same <cardinality> as the real-sequence space because $\mathbb Q$ is countable. The previous part gives the <upper bound> $2^{\aleph_0}$, while <constant functions> give the matching lower bound. Hence
$$
\boxed{|C(\mathbb R,\mathbb R)|=2^{\aleph_0}.}
$$
This computes the <cardinality of continuous real functions>. All five <sets> consequently have the same <cardinality>. The <continuity> assumption is decisive for restriction to a <dense subset> to determine a <function>.
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