I was unable to solve this problem asked in my exam of Topology and need help.

True or False : Does A continuous function on $\mathbb{Q} $ Intersection [0, 1] can be extended to a continuous function on [0, 1] .

I couldn't think of what theorem or counterexample I should use and hence I am posting here.

Any help please.

  • $\begingroup$ Are you asking if a continuous function $f$ on $\mathbb{Q} \cap [0,1]$ can always be extended to a continuous function on $[0,1]$, or if there are continuous functions on this domain that are not continuous on $[0,1]$? $\endgroup$
    – ccroth
    Jun 25, 2020 at 17:40
  • $\begingroup$ @ccroth if a continuous function f on Q∩[0,1] can be extended to a continuous function on [0,1]. Is this statement true or false? $\endgroup$
    – Avenger
    Jun 25, 2020 at 17:44

1 Answer 1


Hint: What is the simplest function you can think of that blows up at $1/\sqrt 2?$

  • 2
    $\begingroup$ I was thinking $1/(x-1/\sqrt 2).$ $\endgroup$
    – zhw.
    Jun 25, 2020 at 18:07
  • $\begingroup$ Can you please help me with this question also if you have some spare time. math.stackexchange.com/questions/3732610/… $\endgroup$
    – Avenger
    Jun 25, 2020 at 22:09

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