3 · Build a function from simple pieces

For each box, an indicator function is one on that box and zero on all the others. Integer multiples of these simple functions can build any integer-valued function on a finite stage.

Try this. Add the basis functions until the filled bars match the target outline. Each label is an integer coefficient, and the final function uses no fractions.

Nöbeling's theorem proves the stronger result for all continuous integer-valued functions on a complete profinite probe. This activity checks only one finite stage, where the basis is elementary.