Continuous only on the Rationals?

Calculus Level 4

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


Note: The following function f:RRf: \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:NQ a: \mathbb{N} \rightarrow \mathbb{Q} .
\hspace{1.0cm} Define f:RR f: \mathbb{R} \rightarrow \mathbb{R} as f(x)=a(n)<x2n.\displaystyle f(x) = \sum_{ a(n) < x } 2^{-n }.

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