A function which is continuous on Z only

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The discussion revolves around finding a function defined on R that is continuous only at the integers (Z). Participants suggest starting with a function that is discontinuous everywhere and modifying it to achieve continuity at the integers. One proposed function is f(x) = x if x is rational and f(x) = -x otherwise, though its correctness is questioned. The conversation also touches on the challenge of ensuring continuity specifically at the origin while maintaining discontinuity elsewhere. The need for clarification and further examples highlights the complexity of the problem.
HappyN
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I have spent ages on this final part of a question but don't seem to be going anywhere - any help would be greatly appreciated!

Given a function f:R->R let X be the set of all points at which f is continuous.
Find an example of a function defined on R which is continuous on Z only.
 
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Hi HappyN! :smile:
HappyN said:
Find an example of a function defined on R which is continuous on Z only.

Find a function continuous at the origin only, and then repeat it. :wink:
 
tiny-tim said:
Find a function continuous at the origin only, and then repeat it. :wink:

Thanks, I'm not quite sure what you mean by 'repeat it' though? If a function is only continuous at the origin surely it is not continuous on all Z?
 
So have you found a function continuous at the origin only?
 
HappyN said:
I have spent ages on this final part of a question but don't seem to be going anywhere - any help would be greatly appreciated!

Given a function f:R->R let X be the set of all points at which f is continuous.
Find an example of a function defined on R which is continuous on Z only.

Start with a function that is discontinuous everywhere and see if you can modify it so that it becomes continuous at the integers but nowhere else.
 
Landau said:
So have you found a function continuous at the origin only?

I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?
 
HappyN said:
I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?

what happens at integers other than zero?
 
HappyN said:
I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?

ok, now chop out a bit round the origin, and keep copy-and-pasting it :wink:
 

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