Power series have identical convergence properties as in R, and thus important transcendental functions exist and behave as in R. Furthermore, there is a natural way to extend any other real function under preservation of its smoothness properties. In addition to these common functions, delta functions can be introduced directly.
A calculus involving continuity, differentiability and integrability is developed. Central concepts like the intermediate value theorem, mean value theorem, and Taylorís theorem with remainder hold under slightly stronger conditions. It is shown that, up to infinitely small errors, derivatives are differential quotients, i.e. slopes of infinitely small secants. While justifying the intuitive concept of derivatives of the fathers of analysis, it also offers a practical way of calculating exact derivatives numerically.
The existence of infinitely small and large numbers allows an introduction of delta functions in a natural way, and the important theorems about delta functions can be shown using the calculus on R.
M. Berz, Lecture Notes, 1992 and 1995 Mathematics Summer Graduate Schools of the German National Merit Foundation, MSUCL-933, Department of Physics and Astronomy, Michigan State University (1994)
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