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Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and = () whenever a and b are coprime.. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields. This avoidance of periodic functions also means that every element in a Hardy field has a (possibly infinite) limit at infinity, so if f is an element of H, then ()
Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including: addition, subtraction, multiplication and division of periodic functions, and; taking a power or a root of a periodic function (provided it is defined for all ).
[6] [7] [8] Operations on functions include composition and convolution. [9] [10] Operations may not be defined for every possible value of its domain. For example, in the real numbers one cannot divide by zero [11] or take square roots of negative numbers. The values for which an operation is defined form a set called its domain of definition ...
In number theory, an arithmetic, arithmetical, or number-theoretic function [1] [2] is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. [3] [4] [5] Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of ...
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