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Find a function defined on $[0,1]$, valued on an interval, but is discontinuous at each point.

That is, try to find a function $f: [0,1]\to \Bbb R$ such that $f([0,1])$ is an interval, but discontinuous at each point.

Here is my try. Let $\Bbb Q=\{r_1,r_2,\cdots\}$ with $r_1=0, r_2=1, \cdots$. Then define $f(x)=x$ for irrational $x$, $f(r_n)=r_{n+1}$, then $f([0,1])=(0,1]$, and is discontinuous at rational points, but how about irrational points?

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Try $$ f(x) = \cases {x & if $x$ is rational\cr x + 1/2 & if $x < 1/2$ and $x$ is irrational\cr x - 1/2 & if $x > 1/2$ and $x$ is irrational}$$

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  • $\begingroup$ What does $x>\frac12$ is irrational even mean?? $\endgroup$ Sep 20 '18 at 23:02
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    $\begingroup$ I mean $x > 1/2$ and $x$ is irrational. $\endgroup$ Sep 20 '18 at 23:03
  • $\begingroup$ lol ............ $\endgroup$ Sep 20 '18 at 23:11

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