Continuous only on the Rationals?

Calculus Level 4

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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