Cauchy's Functional Equation
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Cauchy''s functional equation is one of the simplest functional equations to represent, however its solution over the real numbers is extremely complicated.On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. The fifth problem on Hilbert''s list is a generalisation of this equation.We prove below th...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Cauchy''s functional equation is one of the simplest functional equations to represent, however its solution over the real numbers is extremely complicated.On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. The fifth problem on Hilbert''s list is a generalisation of this equation.We prove below that any other solutions must be highly pathological functions.
