Consider the polynomial,

is you multiply it out you get some negative terms and some positive terms. So we can write (and now has only positive terms).

Example:Say then and .

Now define to be at rational points and at irrational points.

Example:As above define to be at the rational points and at the irrational points.

Then it is continous precisely when so so so . This mean it is continous at exactly these points.

Note: It is not really necessary to do what I did there are many ways to construct such functions but I did it because it looks more pleasing.