  What is the nature - continuous or discrete - of matter and of its
fundamental interactions? The physical meaning, the properties and the
consequences of a discrete scalar field are discussed; limits for the validity
of a mathematical description of fundamental physics in terms of continuum
fields are a natural outcome of discrete fields with discrete interactions. Two
demarcating points (a near and a far) define a domain where no difference
between the discrete and the standard continuum field formalisms can be
experimentally detected. Discrepancies, however, can be observed as a
continuous-interaction is always stronger below the near point and weaker above
the far point than a discrete one. The connections between the discrete scalar
field and gravity from general relativity are discussed. Whereas vacuum
solutions of general relativity can be retrieved from discrete scalar field
solutions, this cannot be extended to solutions in presence of massive sources
as they require a true tensor metric field. Contact is made, on passing, with
the problem of dark matter and the rotation curve of galaxies, with inflation
and the accelerated expansion, the apparent anomaly in the Pioneer spacecraft
acceleration, the quantum Hall effect, high-$T_{c}$ superconductivity, quark
confinement, and with Tsallis generalized one-parameter statistics as some
possible manifestation of discrete interaction and of an essentially discrete
world.
