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This paper introduces the expanded real numbers as an ordered subring of thehyperreal number field that does not contain any infinitesimals, and definesthe set of all integrable functions from the real numbers to the expanded realnumbers. This allows to identify the Dirac delta with a specialhyperreal-valued function of a real variable: the Dirac delta function thusdefined is a general tool, applicable for the mathematical modeling of physicalsystems in which infinitely high densities occur.

Hyperreal delta functions as a new general tool for modeling physical states with infinitely high densities