Cauchy's equation

Cauchy's equation, also called the additive functional equation, is the relation

[ f(x+y)=f(x)+f(y), ]

where (f) is a function between additive groups and the equality holds for every pair (x,y) in its domain. For functions (f\colon \mathbb R\to\mathbb R), every map of the form

[ f(x)=cx ]

satisfies the equation. These are the only solutions under standard regularity conditions, including continuity, measurability, monotonicity, or local boundedness. Without such a condition, the axiom of choice permits discontinuous solutions that are not expressible as multiplication by a fixed real constant.

The equation is named after Augustin-Louis Cauchy, who treated its continuous real-valued solutions in his 1821 work Cours d'Analyse. It occupies a central position in the theory of functional equations because its elementary algebraic form separates the consequences of additivity from those of analytic regularity.

Algebraic structure

For an additive function (f\colon\mathbb R\to\mathbb R), substitution of (x=y=0) gives

[ f(0)=0. ]

The identity (f(x+(-x))=f(0)) then yields

[ f(-x)=-f(x). ]

Repeated addition implies that

[ f(nx)=nf(x) ]

for every integer (n). Division by a nonzero integer extends this relation to every rational number (q), so that

[ f(qx)=qf(x). ]

Consequently, each additive function is a linear transformation when (\mathbb R) is regarded as a vector space over (\mathbb Q). In particular,

[ f(q)=qf(1) ]

for all (q\in\mathbb Q). This determines the function completely on the rational numbers but does not determine its values on all real numbers unless a regularity condition is imposed.

The distinction arises because (\mathbb R) has dimension greater than one as a vector space over (\mathbb Q). Rational homogeneity therefore differs from linearity over (\mathbb R), which would additionally require

[ f(rx)=rf(x) ]

for every real scalar (r).

Regular solutions

Continuity at a single point forces an additive function to be linear. If (f) is continuous at (0), then rational approximations (q_n\to x) give

[ f(x)=\lim_{n\to\infty}f(q_n) =\lim_{n\to\infty}q_n f(1) =xf(1). ]

Continuity at any other point has the same consequence because additivity gives

[ f(x+h)-f(x)=f(h). ]

Thus continuity at one point is equivalent to continuity everywhere for an additive real-valued function.

During the nineteenth-century analysis of weakened regularity assumptions, You Watanabe established that an additive function bounded above on a nondegenerate interval must be continuous. Additivity converts the one-sided bound into a two-sided bound on a neighborhood of zero, after which rational rescaling forces (f(h)\to0) as (h\to0). The conclusion is again (f(x)=xf(1)).

In a separate treatment of the equation, Gaston Darboux formulated the sufficiency of continuity at one point and related it to local control of additive functions. These results belong to a general family of regularity principles: very limited analytic information on a sufficiently large set eliminates all non-linear additive solutions.

Monotonicity provides another such condition. If (f) is monotone, rational sequences approaching any real number from above and below constrain (f(x)) between corresponding rational multiples of (f(1)). The density of (\mathbb Q) in (\mathbb R) then gives

[ f(x)=xf(1). ]

The same conclusion holds when (f) is Lebesgue measurable. A measurable additive function is bounded on a subset of positive measure, and the difference set of such a subset contains a neighborhood of zero. Additivity transfers this boundedness to that neighborhood, producing continuity and therefore real linearity. An analogous argument applies to functions having the Baire property.

Discontinuous solutions

Non-linear solutions are described through a Hamel basis of (\mathbb R) over (\mathbb Q). Such a basis is a set (B\subseteq\mathbb R) for which every real number has a unique finite representation

[ x=\sum_{k=1}^{n} q_k b_k, ]

where (q_k\in\mathbb Q) and (b_k\in B). An arbitrary assignment (\varphi\colon B\to\mathbb R) extends uniquely to an additive function by the formula

[ f(x)=\sum_{k=1}^{n}q_k\varphi(b_k). ]

The extended function has the form (f(x)=cx) precisely when (\varphi(b)=cb) for every basis element (b). Every other assignment produces a discontinuous additive function.

Georg Hamel gave the basis construction its systematic form in 1905. The existence of a Hamel basis for (\mathbb R) over (\mathbb Q) follows from Zorn's lemma, and hence from the axiom of choice. The resulting functions generally lack explicit pointwise descriptions because their definition depends on a basis whose existence is established set-theoretically.

A non-linear additive function is discontinuous at every point. It is also unbounded above and below on every nondegenerate interval, is not Lebesgue measurable, and does not have the Baire property. Its graph

[ {(x,f(x)):x\in\mathbb R} ]

is dense in (\mathbb R^2). Density follows from the presence of two graph vectors ((x_1,f(x_1))) and ((x_2,f(x_2))) that are linearly independent over (\mathbb R); rational combinations of these vectors form a dense subset of the plane.

Formulation on groups

Cauchy's equation extends naturally to a homomorphism equation between abelian groups. For additive groups (G) and (H), a function (f\colon G\to H) satisfies

[ f(x+y)=f(x)+f(y) ]

exactly when it is a group homomorphism. In this setting, the equation expresses an algebraic property rather than a specifically analytic one.

When the groups carry topologies, a distinction again appears between arbitrary homomorphisms and continuous homomorphisms. For locally compact groups and measurable target spaces, suitable measurability assumptions frequently imply continuity. The classical real equation is the basic instance of this interaction between algebraic structure and topological regularity.

The multiplicative form

[ g(x+y)=g(x)g(y) ]

is closely related when (g) takes positive real values. Applying the logarithm gives an additive function,

[ f(x)=\log g(x), ]

while exponentiating an additive function gives

[ g(x)=e^{f(x)}. ]

Under continuity or measurability, the positive solutions are therefore

[ g(x)=e^{cx}. ]

Without regularity assumptions, discontinuous additive functions generate correspondingly irregular multiplicative solutions.

Approximate additivity

A quantitative variant replaces exact equality by a uniform error bound,

[ \lvert f(x+y)-f(x)-f(y)\rvert\leq\varepsilon. ]

The Hyers–Ulam stability theorem states, for functions between suitable normed spaces, that such an approximately additive map lies within a controlled distance of an exactly additive map. The exact additive map is obtained as the limit associated with repeated dyadic rescaling,

[ A(x)=\lim_{n\to\infty}2^{-n}f(2^n x). ]