consider $f:[a,b]\to \mathbb{R}$ such that $f(x) \in \mathbb{Q}$ when $x \in \mathbb{R} \setminus \mathbb{Q} \cap [a,b]$ and $f(x) \in \mathbb{R} \setminus \mathbb{Q}$ when $x \in \mathbb{Q} \cap [a,b]$. Then does there exist continuous function of this type ? I need an example ..if not then why?
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Just a function? Take $f(x)=1$ for irrational $x$ and $f(x)=1/\pi$ for rationals. It is not a very interesting function. – uniquesolution Sep 23 '15 at 13:44
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sorry i cant remember the question exactly because the ques is asked in a interview ..sorry for printing mistake @uniquesolution – Rupsa Sep 23 '15 at 13:52
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Sure. Choose your favorite [the second-favorite works just as well] rational number $a$ and irrational numbers $b$, then define $f$ via $$f(x) = \begin{cases} a & x \in \mathbb{R} \setminus \mathbb{Q} \\ b & x \in \mathbb{Q}\end{cases}.$$
Dominik
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