Welcome to our community

Be a part of something great, join today!

Problem of the Week #215 - Jul 12, 2016

Status
Not open for further replies.
  • Thread starter
  • Moderator
  • #1

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
Here is this week's POTW:

-----
Does there exist a real-valued function on $\Bbb R$ that is discontinuous only on the irrationals?

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!
 
  • Thread starter
  • Moderator
  • #2

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
No one answered this week's problem. You can read my solution below.


No. If so an function $f$ existed, then its oscillation $\omega_f$ would be identically zero on $\Bbb Q$. The rationals can then be written as a countable intersection of open sets $A_n := \{x : \omega_f(x) < 1/n\}$. This implies $\Bbb Q$ is a G$_{\delta}$ set, in $\Bbb R$, contradicting the Baire category theorem.
 
Status
Not open for further replies.