The problem of specifying a scaling function is the problem of choosing an appropriate standard.
A standard has to be commensurable with the things being measured. In fact, it has to be an element of the set of things being measured. Thus, if one wishes to measure the shape of functions which are, say, piecewise constant, then the scaling function should have the same property. If the functions to be measured have compact support, then the scaling function should also have compact support. If the functions to be measured have continuous derivatives, then the scaling function better have that property also. Thus the requirement of commensurability dictates the choice of an appropriate standard - an appropriate scaling function.