We have seen how Oresme, and the other mediaeval scholastics defined and studied the intensity of magnitudes and even drew graphs.

However, the originators of Calculus, Galileo, Fermat, Descartes, Barrow, Leibniz and Newton thought in terms of curves rather than functions; in other words they considered those functions which were defined in terms of known functions such as polynomials, trigonometric and logarithmic functions, conic sections, paths of rolling balls etc. Probably Euler was the first to recognize that a function was something that needed a definition. He said: a function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. By this, he probably had in mind to include power series and indefinite integrals among the objects which should be considered as functions.

One consequence of this vagueness about what a function should be was that everything considered as a legitimate function should be continuous, or at least have only finitely many points of discontinuity. For example, when d'Alembert discussed the vibrating string problem, he insisted that the intial condition must be given by an equation, although Euler allowed finitely discontinuous curves.

But while functions defined by power series always led to continuous and indeed differentiable functions, such problems in applied mathematics often led to trignometric series which can converge to discontinuous functions, so it began to be realised that a function was something that really needed a definition. For example, in his work on power series, Lagrange defined a function as follows: One names a function of one or several quantities any mathematical expression in which the quantities enter in any manner whatever, connected or not with other quantities which one regards as having given or constant values, whereas the quantities of the function may take any possible values.

The first widely accepted 'modern' definition of function was given by Fourier (1768-1830) in his Analytic Theory of Heat (1822). He defined: In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given to the abcissa x, there is an equal number of ordinates f(x)....We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as if it were a single quantity.

In spite of such general definitions being in the air, it was widely considered by mathematicians until the middle of the nineteenth Century that all functions that actually occurred in the real world, such as solutions of differential equations, power series or Fourier series, were differentiable almost everywhere. That belief was blown away by the publication by Weierstrass of a natural looking function that was continuous on the real line, but differentiable nowhere.

Weierstrass' example of a continuous function which is differentiable nowhere:

where b < 1 and ab >1 + 3 p/2 . It is easy to show that the terms are continuous and uniformly convergent, so the series is continuous. It requires more work to show that f(x) is not differentiable for any x . Intuitively, note that if you differentiate termwise, you get

in which the coefficients --> infinity. Later more intuitively clear functions were found such as

Here is a biography of Weierstrass.

The following text, from Weierstrass' lectures in Berlin, shows his approach to differentiable functions using e-d arguments.

In contrast to a non-variable magnitude or constant, which can assume only one value, a variable magnitude is defined to be one which cannot only assume several particular values, but infinitely many ones. It may happen that a variable magnitude can assume every possible positive and negative value; then it is called an unrestrictedly variable magnitude. A variable magnitude may also be restrictedly variable and have a lower or upper bound, or both. The values which a variable magnitude can assume may belong to one or several continuous sequences if the variable magnitude can assume all possibIe values between two bounds. The differential calculus deals only with such continuously variable magnitudes.

Two variable magnitudes may be related in such a way that to every definite value of one there corresponds a definite value of the other; then the latter is called a function of the former. This relationship may extend to several variable magnitudes; accordingly one distinguishes functions with one and with several variable magnitudes. If to one value of the one variable magnitude there always corresponds only one value of another, then the latter is called an unambiguous function and singlevalued function of the former. If to one value of the one magnitude there correspond several values of inother, then the latter is called a multi-valued function of the former. The criterion of a function is that the one variable magnitude changes in general by a definite amount as soon as a definite change of the other one is assumed.

If f(x) is a function of x and if x is a definite value, then the function will change to f(x+h) if x changes to x+h; one calls the difference f(x +h) - f(x) the change which the function undergoes by the change of the argument from x to x+h. If it is now possible to determine for h a bound d such that for all values of h which in their absolute value are smaller than d, f(x+h) - f(x) becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. For one says, if the absolute value of a magnitude can become smaller than any arbitrarily chosen magnitude, however, small, then it can become infinitely small.

If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument.

Theorem. If a continuous function of x has, for a definite value x₁ of the argument, a definite value of the function y₁ , and has for another definite value x₂ a definite value of the function y₂ and if y₃ is an arbitrary value between y₁ and y₂ then there must be between x₁ and x₂ at least one value x₃ for which the function assumes the value y₃ .

The following auxiliary theorems serve as proof.

By the middle of the 19th Century, many mathematicians were actively considering the matter of exactly what a real number is. They were no longer content to accept the Greek idea of magnitude or the vague definitions of their predecessors because 'obvious' assumptions often lead to incorrect conclusions. Dedekind wrote a short booklet called Continuity and Irrational Numbers in 1872, from which the following excerpts are taken. He wrote also a popular version called What are and what should be Numbers?

Click here for a biography of Dedekind.

(a) Introduction

My attention was first directed toward the considerations which form the subject of this pamphlet in the autumn of 1858. As professor in the Polytechnic School in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner.

Among these, for example, belongs the above-mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Durège with whom I had a long and lively discussion. Later I explained these views of a scientific basis of arithmetic to a few of my pupils, and here in Braunschweig read a paper upon the subject before the scientific club of professors, but I could not make up my mind to its publication, because in the first place, the presentation did not seem altogether simple, and further, the theory itself had little promise. Nevertheless I had already half determined to select this theme as subject for this occasion, when a few days ago, March 14, by the kindness of the author, the paper Die Elemente der Funktionlehre by E. Heine (Crelle's Journal, Vol. 74) came into my hands and confirmed me in my decision. In the main I fully agree with the substance of this memoir, and indeed I could hardly do otherwise, but I will frankly acknowledge that my own presentation seems to me to be simpler in form and to bring out the vital point more clearly. While writing this preface (March 20, 1872), 1 am just in receipt of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, by G. Cantor (Math. Annalen, Vol. 5), for which I owe the ingenious author my hearty thanks. As I find on a hasty perusal, the axiom given in Section II of that paper, aside from the form of presentation, agrees with what I designate in Section III as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real number as complete in itself.

(b) Continuity of the straight line Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point p corresponds to the rational number a, then, as is well known, the length op is commensurable with the invariable unit of measure used in the construction, i.e., there exists a third length, a so-called common measure, of which these two lengths are integral multiples. But the ancient Greeks already knew and had demonstrated that there are lengths incommensurable with a given unit of length, e.g. the diagonal of the square whose side is the unit of length. If we lay off such a length from the point o upon the line we obtain an end-point which corresponds to no rational number. Since further it can be easily shown that there are infinitely many lengths which are incommensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals.

If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instrument R constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight line.

The previous considerations are so familiar and well known to all that many will regard their repetition quite superfluous. Still I regarded this recapitulation as necessary to prepare properly for the main question. For, the way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes -- which itself is nowhere carefully defined--and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.

That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operation with these numbers must and can be reduced to the laws of operstion of the positive integers, so we must endeavour completely to define irrational numbers by means of rational numbers here. The question only remains how to do this.

(c) Gaps

The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains. By vague remarks upon the unbroken connection in the smallest parts obviously nothing is gained; the problem is to indicate a precise characteristic of continuity that can serve as the basis for valid deductions.

For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people; the majority may find its substance very commonplace. It consists of the following. In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle: If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.

As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle.

(d) Dedekind cuts

From the last remarks it is sufficiently obvious how the discontinuous domain R of rational numbers may be rendered complete so as to form a continuous domain. Earlier it was pointed out that every rational number a effects a separation of the system R into two classes such that every number a₁, of the first class A₁, is less than every number a₂ of the second class A₂; the number a is either the greatest number of the class A₁ or the least number of the class A₂. If now any separation of the system R into two classes A₁, A₂, is given which possesses only this characteristic property that every number a₁ in A₁ is less than every number a₂ in A₂, then for brevity we shall call such a separation a cut and designate it by (A₁, A₂). We can then say that every rational number a produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this cut possesses, besides, the property that either among the numbers of the first class there exists a greatest or among the numbers of the second class a least number. And conversely, if a cut possesses this property, then it is produced by this greatest or least rational number.

But it is easy to show that there exist infinitely many cuts not produced by rational numbers.

[Dedekind went on to call such cuts irrational numbers, and the set of all cuts he called the real numbers.]

(e) Completeness

Besides these properties, however, the domain of real numbers possesses also continuity; i.e., the following theorem is true: If the system of all real numbers breaks up into two classes U₁, U₂ such that every number a₁ of the class U₁ is less than every number a₂ of the class U₂ then there exists one and only one number a by which this separation is produced.

In the paper mentioned in the first section of the text of Dedekind, he defined the real numbers in a different way, namely as the limits of convergent sequences of rationals. But since different sequences can converge to the same limit, he first has to define equivalence classes of convergent sequences, which he does in the following extract from Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Math. Annalen, Vol. 5, (1872).

The rational numbers form the basis for the determination of the further idea of numerical quantity; I will say that they form a domain A (and include the number zero in it). When I speak of a numerical quantity in the extended sense, it is immediately the case that it is presented in the form of a given infinite series of rational numbers

which have the property that the difference a_n+m- a_n becomes infinitely small with increasing n, whatever the positive integer m may be, or in other words that for an arbitrary given (positive, rational) e there is an integer n₀ so that |a_n+m - a_n| < e, when n >= n₀ and m is an arbitrary positive integer. I express the property of this sequence in the following words: the sequence has a definite limit b.

If there is a second series

which has a definite limit b', one finds that the two sequences can be related to each other in one of the following three ways, which are mutually exclusive; either