  The goal of this paper consists of developing a new (more physical and
numerical in comparison with standard and non-standard analysis approaches)
point of view on Calculus with functions assuming infinite and infinitesimal
values. It uses recently introduced infinite and infinitesimal numbers being in
accordance with the principle 'The part is less than the whole' observed in the
physical world around us. These numbers have a strong practical advantage with
respect to traditional approaches: they are representable at a new kind of a
computer - the Infinity Computer - able to work numerically with all of them.
An introduction to the theory of physical and mathematical continuity and
differentiation (including subdifferentials) for functions assuming finite,
infinite, and infinitesimal values over finite, infinite, and infinitesimal
domains is developed in the paper. This theory allows one to work with
derivatives that can assume not only finite but infinite and infinitesimal
values, as well. It is emphasized that the newly introduced notion of the
physical continuity allows one to see the same mathematical object as a
continuous 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 viewed
as a continuous or a discrete in dependence on the instrument of the
observation used by the researcher. Connections between pure mathematical
concepts and their computational realizations are continuously emphasized
through the text. Numerous examples are given.
