# Real Functions for Physics

**Jorge L. deLyra**

* Department of Mathematical Physics,
Physics Institute,
University of São Paulo *

**July 6, 2015**

### Abstract:

A new classification of real functions and other related real objects
defined within a compact interval is proposed. The scope of the
classification includes normal real functions and distributions in the
sense of Schwartz, referred to jointly as ``generalized functions''.
This classification is defined in terms of the behavior of these
generalized functions under the action of a linear low pass-filter,
which can be understood as an integral operator acting in the space of
generalized functions. The classification criterion defines a class of
generalized functions which we will name ``combed functions'', leaving
out a complementary class of ``ragged functions''. While the
classification as combed functions leaves out many pathological objects,
it includes in the same footing such diverse objects as real analytic
functions, the Dirac delta ``function'', and its derivatives of
arbitrarily high orders, as well as many others in between these two
extremes. We argue that the set of combed functions is sufficient for
all the needs of physics, as tools for the description of nature. This
includes the whole of classical physics and all the observable
quantities in quantum mechanics and quantum field theory. The focusing
of attention on this smaller set of generalized functions greatly
simplifies the mathematical arguments needed to deal with them.