Cauchy Functional Equation
From AoPSWiki
The Cauchy Functional Equation is the functional equation 
As with any functional equation, one may attempt to solve this equation for various different choices of the domain of 
.
Rational Case
If 
is a function whose domain and range are the rational numbers (or any subset of 
closed under addition, like 
or 
), the solutions are only the linear functions 
, with 
.
Real Case
If 
is a function whose domain and range are the real numbers, then there exist solutions to the Cauchy Functional Equation other than linear functions. (To prove this requires some form of the Axiom of Choice, e.g., the existence of a Hamel basis for 
over . As a result, these functions are "pathological" -- in particular, it is not possible to write down a formula for any such function, and the graph of any such function is dense in the plane.)
However, there are a variety of simple "regularity conditions" such that if satisfies one of these conditions and the Cauchy Functional Equation, then in must be of the form 
for some 
. Examples of such conditions are that is continuous (or even just continuous at a single point), that is monotonic, or that 
for all 
.
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