Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains

摘要

The goal of this paper consists of developing a new (more physical andnumerical in comparison with standard and non-standard analysis approaches)point of view on Calculus with functions assuming infinite and infinitesimalvalues. It uses recently introduced infinite and infinitesimal numbers being inaccordance with the principle 'The part is less than the whole' observed in thephysical world around us. These numbers have a strong practical advantage withrespect to traditional approaches: they are representable at a new kind of acomputer - the Infinity Computer - able to work numerically with all of them.An introduction to the theory of physical and mathematical continuity anddifferentiation (including subdifferentials) for functions assuming finite,infinite, and infinitesimal values over finite, infinite, and infinitesimaldomains is developed in the paper. This theory allows one to work withderivatives that can assume not only finite but infinite and infinitesimalvalues, as well. It is emphasized that the newly introduced notion of thephysical continuity allows one to see the same mathematical object as acontinuous or a discrete one, in dependence on the wish of the researcher,i.e., as it happens in the physical world where the same object can be viewedas a continuous or a discrete in dependence on the instrument of theobservation used by the researcher. Connections between pure mathematicalconcepts and their computational realizations are continuously emphasizedthrough the text. Numerous examples are given.