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- Cauchy's_functional_equation abstract "Cauchy's functional equation is the functional equationSolutions to this are called additive functions.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely for any arbitrary rational number .Over the real numbers, this is still a family of solutions; however there can exist other solutions that are extremely complicated. Further constraints on f sometimes preclude other solutions, for example: if f is continuous (proven by Cauchy in 1821). This condition was weakened in 1875 by Darboux who showed that it was only necessary for the function to be continuous at one point. if f is monotonic on any interval. if f is bounded on any interval.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. Such functions are sometimes called Hamel functions.The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number such that are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem from 3-D to higher dimensions.".
- Cauchy's_functional_equation wikiPageExternalLink hunt-for-addictive-monster.html.
- Cauchy's_functional_equation wikiPageExternalLink cauchy.pdf.
- Cauchy's_functional_equation wikiPageID "5270898".
- Cauchy's_functional_equation wikiPageRevisionID "540902043".
- Cauchy's_functional_equation hasPhotoCollection Cauchy's_functional_equation.
- Cauchy's_functional_equation subject Category:Arithmetic_functions.
- Cauchy's_functional_equation subject Category:Functional_equations.
- Cauchy's_functional_equation type Abstraction100002137.
- Cauchy's_functional_equation type ArithmeticFunctions.
- Cauchy's_functional_equation type Communication100033020.
- Cauchy's_functional_equation type Equation106669864.
- Cauchy's_functional_equation type Function113783816.
- Cauchy's_functional_equation type FunctionalEquations.
- Cauchy's_functional_equation type MathematicalRelation113783581.
- Cauchy's_functional_equation type MathematicalStatement106732169.
- Cauchy's_functional_equation type Message106598915.
- Cauchy's_functional_equation type Relation100031921.
- Cauchy's_functional_equation type Statement106722453.
- Cauchy's_functional_equation comment "Cauchy's functional equation is the functional equationSolutions to this are called additive functions.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely for any arbitrary rational number .Over the real numbers, this is still a family of solutions; however there can exist other solutions that are extremely complicated.".
- Cauchy's_functional_equation label "Cauchy's functional equation".
- Cauchy's_functional_equation label "Ecuación funcional de Cauchy".
- Cauchy's_functional_equation label "Equazione funzionale di Cauchy".
- Cauchy's_functional_equation label "Équation fonctionnelle de Cauchy".
- Cauchy's_functional_equation label "Функциональное уравнение Коши".
- Cauchy's_functional_equation label "柯西函數方程".
- Cauchy's_functional_equation sameAs Ecuación_funcional_de_Cauchy.
- Cauchy's_functional_equation sameAs Équation_fonctionnelle_de_Cauchy.
- Cauchy's_functional_equation sameAs Equazione_funzionale_di_Cauchy.
- Cauchy's_functional_equation sameAs 코시_함수_방정식.
- Cauchy's_functional_equation sameAs m.0dbvly.
- Cauchy's_functional_equation sameAs Q680611.
- Cauchy's_functional_equation sameAs Q680611.
- Cauchy's_functional_equation sameAs Cauchy's_functional_equation.
- Cauchy's_functional_equation wasDerivedFrom Cauchy's_functional_equation?oldid=540902043.
- Cauchy's_functional_equation isPrimaryTopicOf Cauchy's_functional_equation.