A function which is continuous on Z only

[HappyN]

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.

 

[tiny-tim]

Hi HappyN!

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.

 

[HappyN]

Find a function continuous at the origin only, and then repeat it.

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?

 

[Landau]

So have you found a function continuous at the origin only?

 

[lavinia]

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.

 

[HappyN]

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?

 

[lavinia]

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

what happens at integers other than zero?

 

[tiny-tim]

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