The infinitesimal instants are non-zero-it is infinitely small.
Infinitesimals have a colorful history.
In the later part of the 18th-century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function.
With the abandonment of infinitesimals in the 19th century, this definition came to be replaced by one employing the more precise concept of limit.
Traditionally, an infinitesimal quantity is one which, while not necessarily coinciding with zero, is in some sense smaller than any finite quantity.
For engineers, an infinitesimal is a quantity so small that its square and all higher powers can be neglected.
In the theory of limits, the term “infinitesimal” is sometimes applied to any sequence whose limit is zero.
An infinitesimal magnitude may be regarded as what remains after a continuum has been subjected to an exhaustive analysis, in other words, as a continuum “viewed in the small.” It is in this sense that continuous curves have sometimes been held to be “composed” of infinitesimal straight lines.
A major development in the re-founding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis (SIA).
Based on the ideas of the American mathematician F. W. Lawvere, and employing the methods of category theory, smooth infinitesimal analysis provides an image of the world in which the continuous is an autonomous notion, not explicable in terms of the discrete.
It provides a rigorous framework for mathematical analysis in which every function between spaces is smooth (i.e., differentiable arbitrarily many times, and so in particular continuous) and in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes.
Since in SIA all functions are continuous, it embodies in a striking way Leibniz's principle of continuity Natura non facit saltus.