My Math Forum state an invertible function that is not continuous:
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 February 15th, 2017, 09:56 AM #1 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2 state an invertible function that is not continuous: State an invertible function that is not continuous. Would this function work? f(x) = x if x is not equal to -1 and 1 and f(-1) = 1 and f(1) = -1 since this function is still surjective and injective. Thanks! Last edited by skipjack; February 15th, 2017 at 01:00 PM.
 February 15th, 2017, 10:24 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,590 Thanks: 1434 a much easier example is just $f(x) = \dfrac 1 x$ Thanks from agentredlum
 February 15th, 2017, 01:07 PM #3 Global Moderator   Joined: Dec 2006 Posts: 21,037 Thanks: 2274 With $f$(0) = 0.
February 15th, 2017, 01:52 PM   #4
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Quote:
 Originally Posted by skipjack With $f$(0) = 0.
or just not included in the domain

0 isn't in the range

 February 15th, 2017, 02:11 PM #5 Global Moderator   Joined: Dec 2006 Posts: 21,037 Thanks: 2274 No, because omitting 0 from the domain makes the function continuous.
February 15th, 2017, 03:16 PM   #6
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Quote:
 Originally Posted by romsek a much easier example is just $f(x) = \dfrac 1 x$
This function is continuous.

A strategy for this is to pick your favorite continuous injective function whose domain and range are not $\mathbb{R}$. Now pick any value not in the domain, $x_0$, and another value not in the range, $y_0$ and specify that your new function takes the values of the continuous function but additionally that $f(x_0) = y_0$.

Use this strategy to come up with an explicit example.

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