= Solution
The <set> $D=\mathbb Q+i\mathbb Q$ is <countable> and dense in $\mathbb C$. Restriction to $D$ is <injective> on <continuous functions> $\mathbb C\to\mathbb R$: if two such <functions> agree there, their equality <set> is closed and contains a <dense set>, so is all of $\mathbb C$. Equivalently a <sequence> of points of $D$ converging to any chosen point and <continuity> determine its value uniquely.
Thus the family has <cardinality> at most
$$
|\mathbb R^D|=\mathfrak c^{\aleph_0}
=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}=\mathfrak c.
$$
For the <lower bound>, every <real number> determines a different constant <continuous function>. Cantor-Schröder-Bernstein yields
$$
\boxed{|C(\mathbb C,\mathbb R)|=\mathfrak c.}
$$
The reduction to countably many values is the feature that separates this answer from unrestricted <functions>.
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