= Cauchy's functional equation
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{title2=$f(x+y)=f(x)+f(y)$}
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= Cauchy functional equation
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<Cauchy's functional equation> asks for <functions> $f:\mathbb R\to\mathbb R$ satisfying
$$
f(x+y)=f(x)+f(y)\qquad(x,y\in\mathbb R).
$$
The identity must hold for every pair of <real numbers>, and a solution is an <additive function>. It says that combining two inputs before applying $f$ gives the same result as combining their outputs.<Remarks on the Cauchy functional equation and variations of it by Daniel Reem>[Daniel Reem, Remarks on the Cauchy functional equation and variations of it, version 7, §1.1, Eq. (1) and the real domain]{ref}
For example, $f(x)=3x$ satisfies the identity because $3(x+y)=3x+3y$. At $x=2/3$ and $y=5/6$ the two sides give
$$
f\left(\frac23+\frac56\right)=f\left(\frac32\right)=\frac92,
\qquad
f\left(\frac23\right)+f\left(\frac56\right)=2+\frac52=\frac92.
$$
For positive widths, a <rectangle> of height three makes this visible: cutting its width into $x$ and $y$ partitions its <area> into $3x$ and $3y$. The same identity also holds for negative inputs, where $3x$ is a signed output rather than an ordinary <area>.
\Image[/cauchy-additivity-rectangles.png]
{title=The function f(x)=3x preserves addition: partitioning a rectangle and checking exact rational grid sums}
Every <function> $f(x)=cx$ with a fixed <real number> $c$ solves <Cauchy's functional equation>. The surprising question is whether these are all the solutions. <Rational homogeneity of an additive function> forces $f(q)=qf(1)$ for every <rational number> $q$, but reaching an <irrational number> requires an additional argument. <Continuity> at just one point or being a <bounded function> on an <interval> of positive length supplies that argument; without such a condition, <discontinuous additive functions> exist.<Remarks on the Cauchy functional equation and variations of it by Daniel Reem>[Reem, version 7, §1.1, printed p. 2: classical continuity and bounded-interval conditions and nonlinear solutions]{ref}
This makes <Cauchy's functional equation> a useful test of the boundary between <algebra> and <analysis>: a rule about <addition> fixes all rational multiples, while a tiny amount of <continuity> fixes the entire <real line>. For example, the <continuous exponential functional equation> reduces to an <additive function> after taking a <logarithm>, so this result explains why its nonzero solutions are <exponential functions>.
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