A function that adds and is nowhere a line

The choice nobody can write down listed the places where the axiom of choice hides, and one line of the list was brief: every vector space has a basis. For the real numbers regarded as a vector space over the rationals, that basis exists only because the axiom says it does. This essay is about what the basis builds — a function that adds correctly and looks like nothing a function is supposed to look like.

The equation is Cauchy’s, from 1821:

$$f(x + y) = f(x) + f(y) \quad \text{for all real } x, y.$$

Its obvious solutions are the straight lines through the origin, $f(x) = kx$. Cauchy proved that if $f$ is continuous there are no others. Without continuity there are, and the figure below is one of them, as much of it as can be drawn exactly.

What adding forces #

Start with what the equation settles on its own. Put $x = y = 0$ and $f(0) = 2f(0)$, so $f(0) = 0$. Adding $1$ to itself $n$ times gives $f(n) = n\,f(1)$. Splitting $1$ into $q$ equal parts gives $q\,f(1/q) = f(1)$, so $f(1/q) = f(1)/q$, and combining the two, $f(p/q) = (p/q)\,f(1)$.

So an additive function is completely determined at every rational number by the one number $f(1)$, and there it agrees with the line $y = f(1)\,x$.

The figure is a picture of how much and how little the equation knows. The rational points are dense — every interval contains some — so they look like a line. But they are countable, and the gaps between them contain almost every real number. At an irrational point, additivity says nothing unless something else ties the function down: continuity, or being bounded on some interval, or being monotone, or even being measurable in Lebesgue’s sense. Any one of those forces $f(x) = f(1)\,x$ everywhere, because each lets the values at nearby rationals control the value between them.

Without any of them, the values at irrational numbers are free — subject only to additivity itself — and the question is how free.

A plane squeezed onto a line #

The freedom lives in how the real numbers are built out of the rationals, and a small piece of it can be seen exactly.

Consider only the numbers of the form $a + b\sqrt2$ with $a$ and $b$ rational. Adding two of them gives another, so they form a collection closed under addition and under multiplication by rationals — a vector space over the rationals — and it is two-dimensional, with $1$ and $\sqrt2$ as a basis. The two coordinates are genuinely independent: if $a + b\sqrt2 = 0$ with $b \ne 0$, then $\sqrt2 = -a/b$ would be rational. So every such number has exactly one pair of coordinates.

That is a two-dimensional grid laid onto a one-dimensional line without any two of its points landing together. With rational coordinates instead of whole ones, the points fill the line densely. The squashing is what makes the wild function possible: a map that is perfectly well behaved on the plane of coordinates — “read off the first coordinate” — becomes, on the line, a map that jumps between wildly different values at points arbitrarily close together.

Additive functions on the finite piece #

On the numbers $a + b\sqrt2$, any rule of the form

$$f(a + b\sqrt2) = a + c\,b$$

is additive, whatever number $c$ is, because it is linear in the coordinates. The rule fixes every rational (where $b = 0$) and sends $\sqrt2$ to $c$. Choosing $c$ is choosing where $\sqrt2$ goes, and additivity allows any choice.

Only one choice gives a straight line.

The three pictures are the whole phenomenon in miniature. Every choice of $c$ other than $\sqrt2$ produces an additive function whose graph is dense in the plane: the points $(a + b\sqrt2,\, a + cb)$ are the image of the rational plane under a linear map that is invertible when $c \ne \sqrt2$, and the image of a dense set under an invertible linear map is dense. The choice $c = \sqrt2$ is the unique one where the map collapses the plane onto a line. Additivity with the wrong value at one irrational number is enough to shatter the graph over every interval.

Adding is not the same as scaling #

It is worth pinning down why an additive function is not automatically linear, because the words suggest it should be.

A linear map, in the sense of the maps that act on a grid, has to respect two operations: adding vectors, and multiplying them by scalars. Cauchy’s equation asks only for the first. From it, as the rational figure showed, the second follows for rational scalars: $f(qx) = q\,f(x)$ whenever $q$ is rational, by adding and splitting. So an additive function is exactly a map that is linear over the rationals.

What it need not do is respect multiplication by an irrational scalar, and the functions drawn above do not: $f(\sqrt2 \cdot 1) = c$, while $\sqrt2 \cdot f(1) = \sqrt2$, and the two agree only when $c = \sqrt2$. Over the real numbers as scalars, the line is one-dimensional and every linear map is $x \mapsto kx$. Over the rationals as scalars, the same line is a vector space of uncountable dimension, and linear maps on it are as varied as the choices of values on its basis.

In finite dimensions over the real numbers, linear maps are automatically continuous — a matrix cannot tear space — and it is tempting to carry that intuition over. It does not carry. Automatic continuity is a fact about finite-dimensional spaces over complete fields, and the reals over the rationals are neither finite-dimensional nor over a complete field. The wild functions are what that failure looks like.

From a finite piece to every real number #

The functions above are defined only on numbers of the form $a + b\sqrt2$. To get an additive function on all real numbers, the same recipe needs a set of real numbers playing the part of $\{1, \sqrt2\}$ for the whole line: a set $B$ such that every real number is, in exactly one way, a finite sum of rational multiples of members of $B$. That is a basis of the reals over the rationals, now called a Hamel basis after Georg Hamel, who used one in 1905 to solve Cauchy’s equation completely.

Given such a basis, an additive function is just a choice of value at each basis element, extended by the coordinates — exactly as $c$ was the value at $\sqrt2$ above. Choose the values in proportion to the basis elements and the function is a line. Choose them any other way and the graph is dense in the plane.

The difficulty is the basis itself. It cannot be countable, because a countable basis would make the reals a countable union of countable sets of coordinate combinations, and the reals are uncountable. It must contain uncountably many real numbers, and there is no rule that picks them: no two of its members can be rational multiples of each other, so from every family of mutually rational multiples it keeps at most one — and which one, if any, is decided by no rule at all. Its existence is proved by Zorn’s lemma — take a maximal set of rationally independent reals, which exists by the lemma, and check that maximality forces it to span — and Zorn’s lemma is the axiom of choice.

How many there are #

Once a Hamel basis is fixed, the additive functions are in exact correspondence with the ways of assigning a real value to each basis element, and that makes them easy to count.

The basis has as many elements as there are real numbers — it must, since the reals are its finite rational combinations and a smaller basis would give too few combinations. So an additive function is a choice of one real number for each of continuum-many basis elements, and there are $\mathfrak{c}^{\mathfrak{c}} = 2^{\mathfrak{c}}$ such choices. The continuous ones, the lines $y = kx$, are determined by a single number $k$, so there are only $\mathfrak{c}$ of them. Almost all additive functions are wild, in the strongest sense counting allows: the lines are a set of strictly smaller size inside the whole, by the same diagonal gap that separates a set from the set of its subsets.

That is a common shape in this part of mathematics. The objects that can be described — continuous functions, measurable sets, definable reals — are few, in the sense of cardinality, and the objects the axiom of choice supplies are almost all of them. The tame ones are the only ones anybody ever meets, because meeting one means describing it.

A graph that is dense and still a function #

The wild graph has a property that sounds contradictory until it is stated carefully. Every vertical line meets it exactly once, because it is the graph of a function. And every disc in the plane, however small and wherever placed, contains points of it. A set can do both: the rationals are a set that every interval meets and that has one point at each rational position, and the wild graph is that kind of set in two dimensions.

It is worth setting beside the other famous monsters of analysis, because it is a different kind. The curve with a corner at every point is continuous — its graph is a genuine curve that can be approximated as closely as desired by drawings — and it fails only to be smooth. Dirichlet’s function, one on the rationals and zero elsewhere, is discontinuous everywhere but perfectly definable. The wild additive function is worse than both in one respect and better in another: it obeys an algebraic law exactly, which neither of the others does, and it cannot be defined at all.

That combination — exact algebraic structure, no analytic control, no definition — is what makes it the standard test case. An argument about functions that is secretly assuming continuity, or measurability, or definability will usually fail on it, and textbooks that claim “additive implies linear” without a hypothesis are corrected by it.

What the wild functions cannot be #

The wild additive functions have a long list of properties they cannot have, and the list is instructive because every item is a way of being tame.

They are unbounded on every interval: if an additive function were bounded on any interval, however small, it would be a line. They are not monotone on any interval, for the same reason. They are not Lebesgue measurable: a theorem of Banach and of Sierpiński from 1920 says that a measurable additive function is linear. So a wild additive function is a non-measurable object of exactly the kind the set with no size at all is — and indeed from a wild additive function one can build a non-measurable set, and from a Hamel basis one can build both.

That last point locates them precisely in the landscape of the axiom. Solovay showed in 1970 that there is a consistent version of set theory, with a weak form of choice enough for ordinary analysis, in which every set of reals is Lebesgue measurable. In that world every additive function is measurable, hence linear. So “there is an additive function that is not a line” is a statement that ordinary mathematics can neither prove nor refute without taking a side on the axiom of choice: it holds in the worlds with a Hamel basis and fails in Solovay’s, two worlds that both obey the rules.

Where the finite pieces are genuinely used #

It would be easy to conclude that these functions are curiosities. The finite pieces, at least, are working tools, and one of them settled a famous problem.

Hilbert’s third problem asked whether two polyhedra of equal volume can always be cut into finitely many pieces and reassembled into each other, as two polygons of equal area always can. Max Dehn answered no in 1900, with an invariant built from exactly the kind of function drawn above: a map from real numbers to a rational vector space that is additive, fixes the rational multiples of $\pi$ at zero, and is otherwise free. Applied to the dihedral angles of a polyhedron, weighted by edge lengths, it produces a quantity that cutting and regluing cannot change and that differs between a cube and a regular tetrahedron of the same volume — so a dissection between them never comes apart.

Dehn’s argument needs the function only on the finitely many angles of the two polyhedra and their rational combinations — a finite-dimensional piece, like the numbers $a + b\sqrt2$ — so no choice is required: the basis of a finite-dimensional rational space can be found by hand. The full Hamel basis is the same idea taken to the whole real line, where hand-finding stops.

What the pictures cannot show #

Any function on all the reals. Every figure is defined on numbers of the form $a + b\sqrt2$ with small denominators — a countable set, and a tiny finite part of it. The functions on the whole line that the axiom of choice provides cannot be drawn even in principle: any procedure that drew one would be a definition of one, and in Solovay’s model there are none to define.

That the graph is dense. The scatter meets every cell of a six-by-six grid over the window, which is checked; that it meets every open set of the plane is the argument about invertible linear images of the rational plane, not something a finite scatter can show. With small denominators the scatter even shows visible stripes, the rows of equal $a$, which are an artefact of which points were plotted.

Additivity everywhere. The figures check additivity where they compute it, on pairs of plotted points. On the numbers $a + b\sqrt2$ it holds exactly because the function is linear in exact rational coordinates; the plotted positions are floating-point, but the function is not computed from them.

Still open: how much choice is really needed #

The existence of a wild additive function follows from a Hamel basis, which follows from the axiom of choice. The converse direction is subtler. A wild additive function does not give back the full axiom, and it is not known exactly which weak form of choice is equivalent to the existence of a discontinuous solution of Cauchy’s equation; the statement sits somewhere strictly between plain set theory, which cannot prove it, and full choice, which proves it easily. Several natural candidates — the existence of a non-measurable set, the existence of a Hamel basis itself — are known to be different statements, and how they are ordered is the kind of question that set theorists settle one model at a time.

There is also a question that only sounds philosophical. A wild additive function’s values at the irrational numbers are “chosen” in a sense that no description can capture. Whether objects that no description can capture should count as existing is exactly what the axiom of choice decides by fiat, and what the essay on infinitely many guessers makes vivid with a puzzle whose solution works, and cannot be carried out.

An equation that almost forces a line #

Cauchy’s equation forces a function’s values at every rational number and at nothing else. Continuity, or any of several weaker tameness conditions, fills in the gaps with the obvious line. Without them the gaps are filled by choosing values on a basis, and every choice but the obvious one shatters the graph over the whole plane.

The finite pieces of that construction are perfectly concrete — they are drawn above, and Dehn used one to show a cube cannot be cut into a tetrahedron. The complete construction needs a basis for the real numbers over the rationals, and that basis exists only by the axiom of choice. It is the cleanest example there is of a mathematical object that adds up correctly, whose existence is proved, and of which not one can be written down.