Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is **continuous** on *exactly* the rational numbers?

Note: The following function $f: \mathbb{R} \rightarrow \mathbb{R}$ is **discontinuous** on exactly the rational numbers:

$\hspace{1.0cm}$ Index all the rational numbers using the bijection function $a: \mathbb{N} \rightarrow \mathbb{Q}$.

$\hspace{1.0cm}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ as $\displaystyle f(x) = \sum_{ a(n) < x } 2^{-n }.$

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