Does continuous extension exists under specific conditions

Question

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.

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 $[0,1]$? — ccroth, Jun 25, 2020 at 17:40
@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? — Avenger, Jun 25, 2020 at 17:44

zhw. · Accepted Answer · 2020-06-25 17:45:54Z

2

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

answered Jun 25, 2020 at 17:36

zhw.

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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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