Cauchy Functional Equation
The Cauchy Functional Equation refers to the functional equation , with
for all
.
Rational Case
If (or any subset closed to addition like
or
), the solutions are only the functions
, with
.
Real Case
If , then we need a suplementar condition like
continous, or
monotonic, or
for all
, to get that all the solutions are of the form
, with
.
There have been given examples of real functions that fulfill the Cauchy Functional Equation, but are not linear, which use advanced knowledge of real analysis.
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