  Scaling-invariant functions preserve the order of points when the points are
scaled by the same positive scalar (with respect to a unique reference point).
Composites of strictly monotonic functions with positively homogeneous
functions are scaling-invariant with respect to zero. We prove in this paper
that the reverse is true for large classes of scaling-invariant functions.
Specifically, we give necessary and sufficient conditions for scaling-invariant
functions to be composites of a strictly monotonic function with a positively
homogeneous function. We also study sublevel sets of scaling-invariant
functions generalizing well-known properties of positively homogeneous
functions.
  Keywords: scaling-invariant function, positively homogeneous function,
compact level set
