Continuity of Function $f:R-R$

Question

Is there any continous function $f:R-R$ such that $f(x)\in Q$ for all $x\in R/Q$ and $f(x) \in R/Q$ for all $x \in Q$?
All the functions i considered following given conditions are not continous.How do i know for sure if there are any or not?

Answer 1

Such a function would not be constant.

Pick $a<b$ with $f(a)\ne f(b)$.

Then $f$ takes all values between $f(a)$ and $f(b)$.

There are uncountably many rationals between $f(a)$ and $f(b)$.

There are only countably many rationals...