Negligible function

In mathematics, a negligible function is a function ${\displaystyle \mu (x):\mathbb {N} {\rightarrow }\mathbb {R} }$ such that for every positive integer c there exists an integer N_c such that for all x > N_c,

Equivalently, we may also use the following definition. A function ${\displaystyle \mu (x):\mathbb {N} {\rightarrow }\mathbb {R} }$ is negligible, if for every positive polynomial poly(·) there exists an integer N_poly > 0 such that for all x > N_poly

The concept of negligibility can find its trace back to sound models of analysis. Though the concepts of "continuity" and "infinitesimal" became important in mathematics during Newton and Leibniz's time (1680s), they were not well-defined until the late 1810s. The first reasonably rigorous definition of continuity in mathematical analysis was due to Bernard Bolzano, who wrote in 1817 the modern definition of continuity. Later Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain ${\displaystyle \mathbb {R} }$):

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