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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 $f(x) = \dfrac 1 x$ 
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 
Senior Member Joined: Sep 2015 From: USA Posts: 2,590 Thanks: 1434  
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 
Senior Member Joined: Sep 2016 From: USA Posts: 670 Thanks: 440 Math Focus: Dynamical systems, analytic function theory, numerics  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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continuous, function, invertible, state 
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