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?


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.