that solves this equation is called an additive function.
Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely
Over the real numbers, the family of linear maps
an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated.
However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions.
is linear if: On the other hand, if no further conditions are imposed on
then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation.
[1] The fifth problem on Hilbert's list is a generalisation of this equation.
Functions where there exists a real number
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 3D to higher dimensions.
A simple argument, involving only elementary algebra, demonstrates that the set of additive maps
are vector spaces over an extension field of
, so, putting things together, We prove below that any other solutions must be highly pathological functions.
In particular, it is shown that any other solution must have the property that its graph
that is, that any disk in the plane (however small) contains a point from the graph.
From this it is easy to prove the various conditions given in the introductory paragraph.
satisfies the Cauchy functional equation on the interval
, but is not linear, then its graph is dense on the strip
satisfies the Cauchy functional equation on
This shows that only linear solutions are permitted when the domain of
However, as we will demonstrate below, highly pathological solutions can be found for functions
based on these linear solutions, by viewing the reals as a vector space over the field of rational numbers.
Note, however, that this method is nonconstructive, relying as it does on the existence of a (Hamel) basis for any vector space, a statement proved using Zorn's lemma.
(In fact, the existence of a basis for every vector space is logically equivalent to the axiom of choice.)
There exist models such as the Solovay model where all sets of reals are measurable which are consistent with ZF + DC, and therein all solutions are linear.
can be written down, the pathological solutions defined below likewise cannot be expressed explicitly.
can be expressed as a unique (finite) linear combination of the
is a solution to Cauchy's functional equation given a definition of
In particular, the solutions of the functional equation are linear if and only if
Thus, in a sense, despite the inability to exhibit a nonlinear solution, "most" (in the sense of cardinality[4]) solutions to the Cauchy functional equation are actually nonlinear and pathological.