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Difference between revisions of "Cauchy Functional Equation"

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The Cauchy Functional Equation refers to the functional equation <math>f:A\to B</math>, with <cmath> f(x+y) = f(x) + f(y) , </cmath> for all <math>x,y \in A</math>.  
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The '''Cauchy Functional Equation''' is the [[functional equation]] <cmath> f(x+y) = f(x) + f(y)</cmath>.
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As with any functional equation, one may attempt to solve this equation for various different choices of the domain of <math>f</math>.
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==Rational Case==
 
==Rational Case==
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If <math>A=B=\mathbb Q</math> (or any subset closed to addition like <math>\mathbb Z</math> or <math>\mathbb N</math>), the solutions are only the functions <math>f(x)=ax</math>, with <math>a\in\mathbb Q</math>.  
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If <math>f: \mathbb Q \to \mathbb Q</math> is a function whose [[domain]] and [[range]] are the [[rational numbers]] (or any [[subset]] of <math>\mathbb Q</math> [[closed]] under addition, like <math>\mathbb Z</math> or <math>\mathbb N</math>), the solutions are only the [[linear function]]s <math>f(x)=ax</math>, with <math>a\in\mathbb Q</math>.  
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==Real Case==
 
==Real Case==
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If <math>A=B=\mathbb R</math>, then we need a suplementar condition like <math>f</math> continous, or <math>f</math> monotonic, or <math>f(x)>0</math> for all <math>x>0</math>, to get that all the solutions are of the form <math>f(x)=ax</math>, with <math>a\in\mathbb R</math>.
 
  
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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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If <math>f:\mathbb R \to \mathbb R</math> is a function whose domain and range are the [[real number]]s, 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 <math>\mathbb R</math> over <math>\mathbb Q</math>.  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 a function | graph]] of any such function is [[dense]] in the plane.)
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However, there are a variety of simple "regularity conditions" such that if <math>f: \mathbb R \to \mathbb R</math> satisfies one of these conditions and the Cauchy Functional Equation, then in must be of the form <math>f(x) = ax</math> for some <math>a \in \mathbb R</math>.  Examples of such conditions are that <math>f</math> is [[continuous]] (or even just continuous at a single point), that <math>f</math> is monotonic, or that <math>f(x)>0</math> for all <math>x>0</math>.
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Revision as of 12:57, 22 July 2009

The Cauchy Functional Equation is the functional equation \[f(x+y) = f(x) + f(y)\].

As with any functional equation, one may attempt to solve this equation for various different choices of the domain of $f$.


Rational Case

If $f: \mathbb Q \to \mathbb Q$ is a function whose domain and range are the rational numbers (or any subset of $\mathbb Q$ closed under addition, like $\mathbb Z$ or $\mathbb N$), the solutions are only the linear functions $f(x)=ax$, with $a\in\mathbb Q$.


Real Case

If $f:\mathbb R \to \mathbb R$ 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 $\mathbb R$ over $\mathbb Q$. 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 $f: \mathbb R \to \mathbb R$ satisfies one of these conditions and the Cauchy Functional Equation, then in must be of the form $f(x) = ax$ for some $a \in \mathbb R$. Examples of such conditions are that $f$ is continuous (or even just continuous at a single point), that $f$ is monotonic, or that $f(x)>0$ for all $x>0$.


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