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 \(\displaystyle f(x) = \sum_{ a(n) < x } 2^{-n }. \)
by
Deeparaj Bhat