Math 2.0

Difference and functional equations Problem

Cauchy's functional equation

Builds on Continuous functionMonotone functionBasisAxiom of choice

Problem

Find all functions f ⁣:ℝ→ℝf \colon \R \to \R such that f(x+y)=f(x)+f(y)for all x,y∈ℝ.f(x + y) = f(x) + f(y) \quad \text{for all } x, y \in \R.

Remarks

The problem is solved. Every solution satisfies f(0)=0f(0) = 0, f(−x)=−f(x)f(-x) = -f(x) and f(qx)=qf(x)f(qx) = q f(x) for every rational qq, by induction on sums and then division. So f(q)=cqf(q) = cq for all rational qq, where c=f(1)c = f(1). If ff is also continuous, then by density of ℚ\Q in ℝ\R the solution is f(x)=cxf(x) = cx for all real xx. The same conclusion holds under each of these weaker conditions on a solution ff:

  1. ff is continuous at one point;
  2. ff is monotone on some interval;
  3. ff is bounded above on some interval;
  4. ff is measurable for Lebesgue measure.

Without such a condition there are other solutions. Take a basis of ℝ\R as a vector space over ℚ\Q, which exists by the axiom of choice. Every ℚ\Q-linear map ℝ→ℝ\R \to \R solves the equation, and choosing its values on the basis freely gives solutions that are not of the form cxcx. The graph of such a non-linear solution is dense in the plane ℝ2\R^2.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Basis, concept, no evidence levelBasisconceptContinuous function, concept, no evidence levelContinuous functionconceptElement, definition, no evidence levelElementdefinitionMonotone function, concept, no evidence levelMonotone functionconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionChoice function, definition, no evidence levelChoice functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementCauchy's functional equation, problem, no evidence levelCauchy's functional e…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Basis concept
  2. Continuous function concept
  3. Element definition
  4. Monotone function concept
  5. Proposition concept
  6. Set definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Family of sets definition
  11. Ordered pair definition
  12. Union definition
  13. Existential quantifier definition
  14. Universal quantifier definition
  15. Cartesian product definition
  16. Union of a family of sets definition
  17. Subset definition
  18. Uniqueness quantifier definition
  19. Binary relation definition
  20. Function definition
  21. Choice function definition
  22. Axiom of choice statement
  23. Cauchy's functional equation problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
About this page
TagZBMEVWN100
URIm2://ZBMEVWN100
Kindproblem
WritingW0 stub
MSC39B22, 39B52
StatusThis copy of the site has no build data. Run m2 status ZBMEVWN100.
AboutFunction
UsesContinuous function, Monotone function, Basis, Axiom of choice
Changed2 November 2026

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