Fundamental Limitations on Precision

Perhaps one of the most basic ways to argue the case starts from the fundamental fact that all physical measurements of quantities that vary almost continuously are necessarily made within finite and non-zero errors. The representation of nature provided by physics is always an approximate one. All physical measurements, as well as all theoretical calculations related to them, of quantities which are represented by continuous variables, can only be performed with finite amounts of precision, that is, within finite and non-zero errors.

In fact, not only this is true in practice, both experimentally and theoretically, but with the advent of relativistic quantum mechanics and quantum field theory, it became a limitation in principle as well. This is so because in its most fundamental aspect all particles in nature are represented by fields, in the sense that the particles are just energetic excitations of these fields. For finite particle energies the wavelengths of these fields are finite and non-zero. Since it is not possible to localize the particles with an accuracy that falls below the length scales defined by those wavelengths, it follows that it is never possible to measure the position of particles, or the dimensions of objects constructed out of these particles, with mathematically perfect precision.

The use of real functions in physics is meant to be for the representation of relations between physical variables that vary almost continuously. If we assume that the physical quantities are observable, then they can at best be measured or prepared (that is, set) within finite and non-zero errors. A relation such as $y=f(x)$ where both $x$ and $y$ are observable means that a finite-spike discontinuity in the function cannot have any physical meaning. In order to detect such a spike in $y$ it would be necessary to measure or prepare $x$ with infinite precision, which is never possible, not only in practice but in principle. Therefore, there is no possible physical meaning to ragged functions containing any such finite-spike discontinuities.