It evaluates to 1 if a number is an integer and something close to 0 if a number is not.
So you can see the transition from near 0 to 1 is very close to the integer in question.
I think this function could be useful in a lot of situations but here is just one example. Suppose you want to know how many integer points there are on a circle on a grid like so:
This is just plugging the equation for the circle into the function I described, and it outputs:
12 + 3*10^-30
So very close to 12.
Indeed there are 12 integer points that lie exactly on the circle: