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Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
In algebra, an additive map, -linear map or additive function is a function that preserves the addition ... this is Cauchy's functional equation.
(+) = (), satisfied by all exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions = + (), satisfied by all logarithmic functions and, over coprime integer arguments, additive functions
A function : is called an additive function if it satisfies Cauchy's functional equation: (+) = + (),. For example, every map of form x ↦ c x , {\displaystyle x\mapsto cx,} where c ∈ R {\displaystyle c\in \mathbb {R} } is some constant, is additive (in fact, it is linear and continuous).
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: [1] = + ().
In 1978, Themistocles M. Rassias succeeded in extending the Hyers' theorem by considering an unbounded Cauchy difference. He was the first to prove the stability of the linear mapping in Banach spaces. In 1950, T. Aoki had provided a proof of a special case of the Rassias' result when the given function is additive.
In additive number theory and combinatorics, ... is a constant non-zero function, ... The Cauchy–Davenport theorem, ...
The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution.It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.