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### Continuous function that take irrationals to rationals and vice-versa. [duplicate]

Can someone help me? How can I prove that there isn't an everywhere continuous function $f:\mathbb R \rightarrow \mathbb R$ that transforms every rational into an irrational and vice-versa?
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How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset \... 1answer 401 views ### Nowhere Continuous Function [duplicate] I was reading Dirichlet and Thomae's functions and got interested to know about functions which are continuous nowhere. Since these have a lot to do with rationals and irrationals, the next question ... 2answers 115 views ###$f(x)\in \mathbb Q,$if$x\not\in \mathbb Q$and$f(x)\not\in \mathbb Q$, if$x\in \mathbb Q$. [duplicate]$f(x)\in \mathbb Q,$if$x\not\in \mathbb Q$and$f(x)\not\in \mathbb Q$, if$x\in \mathbb Q$. Can$f$be continuous? I have tried using the sequential definition of continuity on rational and ... 1answer 90 views ### There exists a continuous function$f$such that$f(\Bbb Q) \subseteq \Bbb R\setminus\Bbb Q$and$f(\Bbb R\setminus\Bbb Q)\subseteq\Bbb Q$[duplicate] True or false: There exists a continuous function$f: \Bbb R \to \Bbb R$such that$f(\Bbb Q) \subseteq {\Bbb R}\setminus {\Bbb Q}$and$f({\Bbb R}\setminus {\Bbb Q}) \subseteq {\Bbb Q}$. My attempt:... 2answers 69 views ### Continuous Function taking rationals to irrationals and vice versa [duplicate] Note Do not close this question like It was done earlier. Question is different so I am asking a new question. l am not supposed to use connectedness here Mine is a basic real analysis course. ... 1answer 102 views ### Does this type of functions exist [duplicate] Does there exist a cont function$f$:$\mathbb{R}\rightarrow\mathbb{R}$which takes irrational values at rational points and rational values at irrational points? 1answer 110 views ### Existence of Continuous function [duplicate] Does there exist a continuous function$f : \Bbb R \to \Bbb R$which takes irrational values at rational points and rational values at irrational points? 1answer 39 views ### A single-variable continuous function that is irrational if and only if its argument is rational [duplicate] I wonder if it is possible to construct a continuous$f : \mathbb R \to \mathbb R$such that for each$x \in \mathbb R$,$f(x)$is irrational if and only if$x$is rational? my attempt Sadly, I ... 1answer 69 views ### Can such a function be continuous? [duplicate] Let$f$be a function from$\Bbb R$to$\Bbb R$such that$f(x)$is rational when$x$is irrational, and$f(x)$is irrational when$x$is rational. Can$f$be continuous? Thanks for your help. 1answer 60 views ### Existence of a continuos function in$[0, 1]$[duplicate] Can exist a continuous function$f:[0, 1]\rightarrow \mathbb{R}$so that$f(x)\in\mathbb{Q}$if$x\in\mathbb{I\cap [0, 1]}$and$f(x)\in\mathbb{I}$if$x\in\mathbb{Q\cap [0, 1]}$? Why yes? why not? 1answer 30 views ### does there exist example of this paticular type of function? [duplicate] consider$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]...
Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$. I suppose this has ...