Abstract
When variable quantities are so related among themselves that, the value of one of them being given, we are able to deduce the values of all the others, we usually consider these various quantities expressed by means of one among them, which then takes the name of the independent variable; and the other quantities, expressed by means of the independent variable, are what we call functions of this variable.
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Notes
- 1.
The concept of the function had been around for a long time. René Descartes (1596–1650) certainly had the idea in mind as he wrote La géométrie in 1637, but the concept had been floating in the air even before his time. However, in the middle of the 18th century, Leonhard Euler (1707–1783) made the function concept central to the calculus, shifting the focus of the discipline from curves to functions. This shift was instrumental in setting the stage for the rigorization of the calculus.
- 2.
The notation \({\varvec{{L}}}x\) is the one Cauchy will use to denote our \(\log {x}\) today. Additionally, \({\varvec{{l}}}x\) is the notation he will use to denote our \(\ln {x}.\) Both will be used extensively later in the text. This notation has been retained to maintain the feel of Cauchy’s original work.
- 3.
Although Cauchy does not make this perfectly clear until later in this lecture, A is always a positive value. So, the function \(A^x\) is well defined.
- 4.
Cauchy’s critically important definition for a continuous function. His imprecise wording here does not clearly distinguish the difference between continuity at a point and uniform continuity over an interval. Debate over this issue goes on to the present day. Fortunately, the subtle concept of uniform continuity does not create a serious issue in most of Cauchy’s work within his Calcul infinitésimal, as he does nearly all of his analysis in this text for well-behaved functions on closed, bounded intervals. It was shown much later in the 19th century that in this situation, a continuous function is also uniformly continuous – a result of the Heine–Borel Theorem, named after Eduard Heine (1821–1881) and Émile Borel (1871–1956). However, Cauchy’s imprecision here will go on to haunt him in proofs of several theorems to follow.
- 5.
Cauchy’s original text uses the phrase solution de continuité here, translated literally as solution of continuity, which is an older term in use at his time meaning points where continuity dissolves, ceases to exist, or disappears. Today we would certainly call this a point of discontinuity. This older phrase has been retained through this translation.
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Cates, D.M. (2019). OF CONTINUOUS AND DISCONTINUOUS FUNCTIONS. GEOMETRIC REPRESENTATION OF CONTINUOUS FUNCTIONS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_2
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