Chapter III
Functions
How mathematicians argued for a hundred years over a single word
MDCCXVIII
I call a function of a variable magnitude a quantity composed in any manner whatsoever of this variable and of constants.
Johann Bernoulli, 1718 (freely translated)
The question
A word you have been using since primary school. But do you know what it means?
At secondary school, functions are everywhere: linear, quadratic, exponential, trigonometric. We draw their graphs, look for their domains, calculate intersections. But before we start, try to answer a simple question. Pick whatever comes to mind first – there is no right or wrong here.
What is a function?
Pick an answer. You can try more than one.
A function is a graph
as Leibniz and the Bernoullis saw itThis is a graph. Is it a function?
THE CIRCLE x² + y² = 1 is a beautiful graph, but it is not a function: the number x = 0 has two values at once, y = 1 and y = −1. The dashed vertical line cuts the circle twice.
So a graph is not the same thing as a function. And the other way round: we will meet functions whose graph cannot be drawn at all.
A function is an algebraic expression
Johann Bernoulli 1718, Euler 1748A torsion spring with a stop: the torque grows with the angle but never exceeds ±4 N·m.
This has not one formula but three. So is it not a function?
IT IS A FUNCTION. Every angle θ corresponds to exactly one torque T(θ). The number of formulas does not matter.
In 1748 Euler still claimed that a function is an “analytic expression”. Seven years later, in 1755, he rewrote his definition, because physics – the vibrating string – had brought him shapes that could not be written with a single formula.
And a little trick question: |x| = √(x²). Is that one formula, or two pieces of straight lines?
A function is a continuous curve
Euler 1748There is a jump here. You cannot draw it without lifting the pencil. So is it not a function?
IT IS A FUNCTION. To the left of zero its value is 0, from zero to the right its value is 1. The filled dot shows where zero itself belongs – every x has exactly one value.
Beware of a trap: when Euler wrote about a “continuous” curve, he meant a curve given by a single expression. A curve made of pieces he called “discontinuous”, even if it could be drawn in one stroke. The spring from the previous choice would have been “discontinuous” for him. Today’s meaning of continuity was only introduced by Bolzano and Cauchy around 1820.
A function is an infinite series
Lagrange 1797f(x) = e−1/x², f(0) = 0
A smooth curve – and its series at zero is nothing but zeros. Is it a function?
IT IS A FUNCTION. In 1797 Lagrange wanted to build the whole of analysis on the idea that every function can be expanded into a power series a0 + a1x + a2x² + … Derivatives would then be just the coefficients.
In 1822 Cauchy presented this function: all its derivatives at zero are zero, so its series is 0 + 0x + 0x² + … = 0. But the function itself is not zero. So a series does not determine a function uniquely.
A function is a relationship between two quantities
Euler 1755x = y²
There is a clear relationship between x and y. So is it a function?
IT IS NOT. For x = 4 we get both y = 2 and y = −2. The relationship is there, but one condition is missing: uniqueness.
In 1755 Euler was already very close. He wrote that when some quantities depend on others in such a way that the former change when the latter change, the former are called functions of the latter. Today, what we get without the uniqueness condition is called a relation. We will come back to that on the chessboard.
A function is a machine
today’s schoolbook pictureThis machine has an input – a number from 1 to 6 – and an output. Put a three into it.
You put in the same number and a different one comes out every time. Is it a function?
IT IS NOT. A machine that gives something different every time for the same input is a die, not a function. A function must always answer the same input in the same way.
The machine picture is not bad, though. Of all the options it comes closest to today’s definition: a function is an assignment. It does not matter whether we give it by a formula, a graph, a table or in words.
The people
A hundred years of disputes, letters and corrections. Had they had a group chat, it would probably have looked like this.
The word function was invented by Leibniz. Then the Bernoullis, Euler, d'Alembert, Lagrange, Fourier, Cauchy and many others took it up – and each of them meant something slightly different by it. The quarrel flared up most over a perfectly ordinary thing: a vibrating string.
Function ❦ group
1673–1921- Leibniz1673–1694
I am trying out the word functio for line segments that “perform a function” for a curve – tangents, normals, ordinates…
- Johann Bernoulli1718
I propose something more general: I call a function of a variable a quantity composed in any manner whatsoever of this variable and of constants.
- Euler1748
Agreed. A function is an analytic expression. And I call a curve continuous when it is given by a single expression. When it is made of pieces of different expressions, it is discontinuous.
- d'Alembert1747
I have solved the vibrating string. The solution must be given by a single analytic expression, otherwise it is not mathematics.
- Euler1749
But I can shape the string by hand at the start, say into a “little roof”! We must allow such shapes, even if they have no single formula.
-
☞ Is this a function? A string plucked up with a finger in the middle.
Today, without hesitation: yes. Every point of the string has one displacement. For d'Alembert, though, it was not admissible – the little roof needs two formulas.
- d'Alembert1750
I reject that. Such “arbitrary curves” do not belong in analysis.
- Daniel Bernoulli1753
Gentlemen, every vibration is a sum of sine waves. An infinite series will do.
- Euler1755
I am revising my definition: when some quantities depend on others in such a way that the former change when the latter change, the former are called functions of the latter. A formula is no longer necessary.
- d'Alembert1754–1757
In the Encyclopédie I explained the limit as early as 1754, in the article Différentiel. The article Fonction only came in 1757. The limit was in the Encyclopédie before the function!
- Lagrange1797
I propose building the whole of analysis on series. We expand every function into a power series, and the derivatives are just its coefficients.
- Fourier1807
And I claim that even an “arbitrary” function, jumps and all, can be written as a series of sines and cosines.
- Cauchy1821–1822
I am introducing proper continuity: a small change in x causes a small change in y. And Monsieur Lagrange – e−1/x² has a series of nothing but zeros at zero, and yet it is not zero. A series does not determine a function.
- Dirichlet1829–1837
It is a function when to each x there corresponds one definite y, however that assignment is given. Example: 1 for rational numbers, 0 for irrational ones. Try drawing that!
- Weierstrass1872
I bring you a function that is continuous everywhere but has a derivative at no point.
- CHHermite1893
I turn away with fright and horror from this lamentable plague – continuous functions without derivatives.
- HPPoincaré1899
New functions used to be invented for some useful purpose. Today they are invented to show the flaws in our fathers’ reasoning. Logic breeds monsters.
- Dedekind1888
I speak in general of a mapping of one set into another. A function is just a special case.
- FHHausdorff and Kuratowski1914–1921
And a mapping is simply a set of ordered pairs. The pair (a, b) can even be defined purely in terms of sets.
The lines are freely retold positions, not verbatim quotations. The dates match the sources.
Who fought on the field of functions?
Eleven men, two centuries, one concept. Follow their path – click a portrait.
Gottfried Wilhelm Leibniz 1646–1716, Leipzig and Hanover
Together with Newton he created the differential calculus. The word functio appears in his manuscripts around 1673 and in print in 1692–1694. His letters with Johann Bernoulli (1694–1698) gave rise to the meaning “a quantity depending on a variable”.
We still use his notation dx and ∫ today.
Johann Bernoulli 1667–1748, Basel
Euler’s teacher. In 1696 he set Europe the brachistochrone problem – the curve of fastest descent. In 1718 he published the first general definition of a function.
He wrote functions with a Greek letter: φx.
Leonhard Euler 1707–1783, Basel, St Petersburg, Berlin
He introduced the notation f(x) (1734, in print 1740). In the Introductio in analysin infinitorum (1748) he treats a function as an analytic expression, in the Institutiones calculi differentialis (1755) already as a dependence between quantities.
He wrote a large part of his work almost blind. Even he needed seven years of thinking about the function.
Jean le Rond d'Alembert 1717–1783, Paris
As a newborn he was left on the steps of the church of Saint-Jean-le-Rond, hence his name. With Diderot he edited the Encyclopédie and was in charge of mathematics and science. In 1747 he derived the wave equation and defended the view that a function is a single expression.
He called the limit “the true metaphysics of the differential calculus”.
Daniel Bernoulli 1700–1782, Basel
Johann’s son. In 1753 he treated the string as a sum of sine waves. He is best known for his work in hydrodynamics (Bernoulli’s equation).
Joseph-Louis Lagrange 1736–1813, Turin, Berlin, Paris
In the book Théorie des fonctions analytiques (1797) he understood a function as a power series. He introduced the notation f′(x).
Joseph Fourier 1768–1830, Auxerre, Grenoble, Paris
He studied the spread of heat and claimed that any function can be written as a series of sines and cosines (1807, book 1822). With that he rekindled the dispute over what an “arbitrary” function is.
He accompanied Napoleon on the expedition to Egypt.
Augustin-Louis Cauchy 1789–1857, Paris
In the Cours d'analyse (1821) he gave precise definitions of limit and continuity. With the function e−1/x² (1822) he showed that Lagrange’s idea of “function = series” does not hold up.
Peter Gustav Lejeune Dirichlet 1805–1859, Berlin, Göttingen
In 1829 he described a function that is continuous at no point. In 1837 he defined a function as any unique assignment.
Karl Weierstrass 1815–1897, Berlin
He taught at a grammar school for a long time before becoming a professor. In 1872 he presented to the Berlin Academy a continuous function without a derivative. It was published in 1875 by his colleague du Bois-Reymond.
Richard Dedekind 1831–1916, Braunschweig
In the book Was sind und was sollen die Zahlen? (1888) he introduced the general concept of a mapping (Abbildung) between arbitrary sets.
Monsters
Once a function need not have a formula, it can be anything. Really anything?
Dirichlet’s definition admitted functions with jumps, with holes, and even functions that cannot be drawn. Try it yourself: for each picture, decide whether it is the graph of a function.
1. Hyperbola
It is. A hole at zero does not mean it is not a function. Zero simply does not belong to the domain.
2. The library desk
It is not. At 11:00 the graph shows two values at once. There are filled dots at both levels.
3. The Dirichlet function
There is deliberately no graph here – try to imagine it.
It is. It assigns exactly one value to every number: 1 to rational ones, 0 to irrational ones. It is a function, even though all it has is a definition. It cannot be drawn: between any two numbers there are both rational and irrational ones, so the “line” at height 1 and at height 0 would be dense everywhere and yet full of holes everywhere. Every picture would lie.
4. Sine of one over x
It is. Near zero it oscillates infinitely fast, but every x ≠ 0 has one value.
Weierstrass’s monster
1872W(x) = Σ (½)n cos(13nπx). Zoom in as far as you like – the curve never becomes smooth. It is continuous, and yet it has a tangent at no point.
“I turn away with fright and horror from this lamentable plague of continuous functions which do not have derivatives.”Charles Hermite in a letter to Stieltjes, 1893
“Logic sometimes breeds monsters.”Henri Poincaré, 1899
The final twist: in 1931 Banach proved that a “typical” continuous function is exactly such a monster. The smooth functions from school are in fact a rare exception.
A function is…
We will finish that sentence on a chessboard.
After all those disputes, mathematics arrived at a surprisingly simple answer. It needs no formula, no graph and no continuity, just pairs and one rule. We will show it on a chessboard.
The chessboard
step 1 of 3Where the mapping came from
- 1718Johann Bernoulli: a function is a quantity composed of a variable and constants.
- 1755Euler: a function is the dependence of one quantity on another.
- 1837Dirichlet: a function is any unique assignment.
- 1888Dedekind introduces the general concept of a mapping (Abbildung).
- 1914Hausdorff: a mapping is a set of ordered pairs.
- 1921Kuratowski defines the ordered pair purely in terms of sets.
- 1939Bourbaki – and thanks to him this definition makes its way into schools.
A FUNCTION IS
an assignment that gives every element x of the domain exactly one element y.
f ⊆ A × B, ∀x ∈ A ∃! y ∈ B: (x, y) ∈ f
What took Euler seven years and mathematics two hundred, we now write on a single line.
And what about your answer from the beginning?
- GraphEvery function has a graph, but not every graph is a function (the circle). Some graphs cannot even be drawn (Dirichlet).
- FormulaA formula is just one of the ways to give an assignment. The spring has three formulas, and yet it is one function.
- Continuous curveContinuity is a property a function may or may not have. And even a continuous function can be a monster (Weierstrass).
- SeriesMany functions can be expanded into a series, but the series does not always determine them (Cauchy).
- RelationshipAlmost right. Only uniqueness was missing: a relationship with it is a function, without it a relation.
- MachineRight, as long as the machine always answers the same input in the same way.
If something in mathematics is not clear to you straight away, never mind. Puzzling over it is part of the process – after all, it took even Euler a few years.
END OF CHAPTER
Sources: A. P. Youschkevitch, The concept of function up to the middle of the 19th century, 1976; I. Kleiner, Evolution of the function concept: a brief survey, 1989; MacTutor History of Mathematics; Encyclopédie (ARTFL, University of Chicago).
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