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I have been doing questions on continuity, and a thought occurred to me, I asked a friend and she said that no such function exists, but I said a function must exist, because it could be constructed precisely so it holds the properties in question, she asked me for an example and I have no idea how to generate one.

Function $f(x)$ on $\mathbb{R}$ that is discontinuous at every point, but $|f(x)|$ is continuous on $\mathbb{R}$

Does such a function exist?

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Yes: let $f(x):=1$ if $x$ is rational and $f(x):=-1$ otherwise.

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  • $\begingroup$ Why does this work? wouldn't there be cases where large amounts of different values are clustered together in the non-rationals? $\endgroup$ – Katie Apr 29 '14 at 22:11
  • $\begingroup$ @Katie, no, the largest value of $f$ is $1$. $\endgroup$ – IAmNoOne Apr 29 '14 at 22:15
  • $\begingroup$ Oh I see now, very, very good, thank you! $\endgroup$ – Katie Apr 29 '14 at 22:16

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