The Gamble Prize (October, 1915) for Mathematics at Girton College, Cambridge, was awarded for this essay to the author, Dr. Grace Chisholm Young.
The matter of the present thesis falls into two parts. In the first part I consider a perfectly general function f(x), not necessarily continuous, and discuss the character of the points at which it has an infinite derivate. In the second part I consider a particular function, the continuous non-differentiable function of Weierstrass.
I have endeavoured, as far as possible, to add such explanations of terms used, and such subsidiary demonstrations, that the paper should be capable of being put into the hands of our more advanced mathematical students, hoping that it may perhaps serve in the future as an introduction to a number of valuable and interesting notions and disciplines, as well as an inducement to original research in a domain which is still to a large extent unexplored.
The rest of the introduction defines the derivate of a function f(x), gives some examples and properties of derivates, and outlines the history of work about differentiable and non-differentiable functions. Young points out that "The earliest publication of a function which has at no point a differential coefficient [a derivative] is due to Weierstrass...It is however by no means certain that Weierstrass was really the first to construct such a function. Professor Raoul Pictet of Geneva informs me that at least as early as 1860, the Swiss mathematician Cellérier, under whom he studied at that time at the University of Geneva, had constructed such a function, and had spoken about it to him."
Young concludes the introduction with an imaginative discourse about an analogy between the study of conic sections and motion of the heavenly bodies, and that of the study of curves with no tangents and the movements of tiny atoms as observed by an ultramicroscope.
"The discovery by Weierstrass and by cellérier of curves with tangents marks indeed an epoch in the history of Mathematics. We of the twentieth century are bound to recognise it in its full importance. It is not only as Du Bois Reymond and a few thinkers of the 19th century pointed out, a remarkable addition to the philosophy of Mathematics. These curves afford us a means of rendering more veracious the representation of the physical universe by the realm of mathematics. I cannot but think that these curves will serve as the basis of the geometrical theory of molecular phenomena, in the same sense as the conic sections have served as a first approximation to the movements of the planets.
Nevertheless the apparent trajectory of an element in the ultramicroscope is not a curve without tangents. We cannot follow all the details; a mental effort is required to coordinate the impressions made on the retina by the sparkling molecules as they pass. But neither is the path of a planet really an ellipse.
At the present moment the reduction of the trajectory of an atom to a question of analytical geometry is not possible. But who knows whether there will not arise a Newton to illuminate our subject? Let us then patiently attack the study of these remarkable curves, leaving to others the work of making the discoveries which may lead to the application of our results in the world of atoms.
In Part I of the essay, Young proves that:
THEOREM: If f(x) is a continuous function the points at which the upper right-hand derivate f+(x) has the value +∞, while the lower derivate on the other side f–(x) is different from –∞, form a set of content zero.
She then gives an example that shows that "a continuous function may have an infinite derivate at any perfect set nowhere dense. Moreover it shows that the fact that one of the derivatives is finite everywhere has no influence on this."
Part II is about "On Weierstrass's non-differentiable function and curves without tangents." Weierstrass's example is the function
f(x) = ∑n=0 bncos(anxπ)
"where a is an odd integer and b a positive quantity less than unity, these two parameters being connected by the inequality ab > 1 + 3π/2 so that in particular a is not less than 7 and b is greater than a quantity which is but little less than 6/a."
The paper concludes with a short appendix on the curve y = (1 – cos x)/x which is used from time to time in the proof of the properties of Weierstrass's function in Part II. This appendix is accessible to any student who has used calculus to analyze the graph of a function to determine where the graph is increasing or decreasing, and to find the locations of the maxima and minima.