Abstract: 
In this paper I go further from the digital continuous axiomatic Euclidean geometry ( [8]) and introduce the basic definitions and derive the basic familiar properties of the differential and integral calculus without the use of the infinite, within finite sets only. No axioms are required in this only successfully chosen definitions. I call it the digital differential and integral calculus. Such mathematics is probably the old unfulfilled hitherto dream of the mathematicians since many centuries. Strictly speaking it is not equivalent to the classical differential and integral calculus which makes use of the infinite (countable and uncountable) and limits. Nevertheless for all practical reasons in the physical and social sciences it gives all the well known applications with a finite ontology which is directly realizable both in the physical ontology of atomic matter or digital ontology of operating systems of computers. Such a digital calculus has aspects simpler than the classical “analogue” calculus which often has a complexity irrelevant to the physical reality. It can become also more complicated than the classical calculus when more than 2 resolutions are utilized, but this complexity is directly relevant to the physical reality. The digital differential and integral calculus is of great value for the applied physical and social sciences as its ontology is directly corresponding to the ontology of computers. It is also a new method of teaching mathematics where there is integrity with what we say, write, see, and think. In this short outline of the basic digital differential and integral calculus, we include on purpose almost only the basic propositions that are almost identical with the corresponding of the classical calculus for reasons of familiarity with their proofs.
