## Evolution of the function concept: A brief survey (1989)

Venue: | The College Mathematics Journal |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Kleiner89evolutionof,

author = {Israel Kleiner},

title = {Evolution of the function concept: A brief survey},

journal = {The College Mathematics Journal},

year = {1989},

volume = {20},

pages = {282--300}

}

### OpenURL

### Abstract

received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the history of mathematics and its use in the teaching of mathematics. Introduction. The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A one-sentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of

### Citations

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Mathematical thought from ancient to modern times
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- 1972
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Citation Context ... need not be such a division of roles. And as long as no special independent role is given to one of the variables involved, the variables are not functions but simply variables [2, p. 348]. See [6], =-=[15]-=-, [27] for details. The calculus developed by Newton and Leibniz had not the form that students see today. In particular, it was not a calculus of functions. The principal objects of study in 17thcent... |

39 |
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- 1984
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Citation Context ...grability of a function in terms of limits. 7 It should be noted that standards of rigor have changed in mathematics (not always from less rigor to more), and that Cauchy’s rigor is not ours. Kitcher =-=[14]-=- suggests that Cauchy’s motivation in rigorizing the basic concepts of the calculus came from work in Fourier series. See also [8] for background to Cauchy’s work in analysis. 7 9s(Bolzano had done mu... |

27 |
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Citation Context ...ro. 14s“using a profound but extremely complex method” [19]. 11 According to Luzin [19], “the impact of Lebesgue’s discovery was just as stunning as that of Fourier in his time.” See [5], [19], [20], =-=[21]-=- for details. Not all functions in the sense of Dirichlet’s conception of function as an arbitrary correspondence are analytically representable (in the sense of Baire), although it is (apparently) ve... |

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Citation Context ...there need not be such a division of roles. And as long as no special independent role is given to one of the variables involved, the variables are not functions but simply variables [2, p. 348]. See =-=[6]-=-, [15], [27] for details. The calculus developed by Newton and Leibniz had not the form that students see today. In particular, it was not a calculus of functions. The principal objects of study in 17... |

9 |
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Citation Context ...y manner whatever of this variable and of constants [23, p. 72]. This was the first formal definition of function, although Bernoulli did not explain what “composed in any manner whatever” meant. See =-=[3]-=-, [6], [12], [27] for details of this section. 2. Euler’s Introductio in Analysin Infinitorum. In the first half of the 18th century, we witness a gradual separation of 17th-century analysis from its ... |

8 |
The Origins of Cauchy’s Rigorous Calculus
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Citation Context ...om less rigor to more), and that Cauchy’s rigor is not ours. Kitcher [14] suggests that Cauchy’s motivation in rigorizing the basic concepts of the calculus came from work in Fourier series. See also =-=[8]-=- for background to Cauchy’s work in analysis. 7 9s(Bolzano had done much of this earlier, but his work went unnoticed for fifty years.) In dealing with continuity, Cauchy addresses himself to Euler’s ... |

8 |
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Citation Context ...h are taken as undefined (primitive) concepts satisfying certain relations or axioms. In fact, in 1966 Lawvere outlined how category theory can replace set theory as a foundation for mathematics. See =-=[11]-=- for details. In the recent developments outlined in this section, we have seen the function concept modified ( L2 functions), generalized (distributions), and finally “generalized out of existence” (... |

6 |
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Citation Context ...ntial equations, seemed to provide a conclusive proof of the fact that functions ‘more general than those expressed by an equation’ were legitimate mathematical objects ... [22, p. 86]. See [3], [4], =-=[9]-=-, [16], [18], [19], [22], [27] for details on section 3. 4. Fourier and Fourier Series. Fourier’s work on heat conduction (submitted to the Paris Academy of Sciences in 1807, but published only in 182... |

6 |
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Citation Context ...not be such a division of roles. And as long as no special independent role is given to one of the variables involved, the variables are not functions but simply variables [2, p. 348]. See [6], [15], =-=[27]-=- for details. The calculus developed by Newton and Leibniz had not the form that students see today. In particular, it was not a calculus of functions. The principal objects of study in 17thcentury ca... |

5 | A source book in classical analysis - Birkhoff - 1973 |

3 |
The concept of function in the 19th and 20th centuries [···]. Archive for History of Exact Sciences
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Citation Context ...cians are free to create their objects at will. There was a famous exchange of letters in 1905 among Baire, Borel, Hadamard, and Lebesgue concerning the current logical state of mathematics (see [5], =-=[20]-=-, [21] for details). Much of the debate was about function theory—the critical question being whether a definition of a mathematical object (say a number or a function), however given, legitimizes the... |

2 |
The development of the concept of function from Euler to Dirichlet
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Citation Context ...tervals. (Thus, f�x� � � was now, for the first time, considered to be a bona fide function.) (b) Functions drawn freehand and possibly not given by any combination of analytic expressions. As Lützen =-=[17]-=- put it: D’Alembert let the concept of function limit the possible initial values, while Euler let the variety of initial values extend the concept of function. We thus see that this extension of the ... |

2 | Some definitions of the concept of function from Joh. Bernoulli to N - Ruthing - 1984 |

1 |
Mathematics and Rational Mechanics;” In: Ferment of Knowledge (eds
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- 1980
(Show Context)
Citation Context ...ction did not originate with Euler, it was he who first gave it prominence by treating the calculus as a formal theory of functions. As we shall see, Euler’s view of functions was soon to evolve. See =-=[2]-=-, [3], [6], [27] for details of the above. 3. The Vibrating-String Controversy. Of crucial importance for the subsequent evolution of the concept of the function was the Vibrating-String Problem: An e... |

1 |
Des Fonctions Comme Expressions Analytiques aux Fonctions Representables Analytiquement
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Citation Context ...efinition of function, was among the first to restrict explicitly the domain of the function to an interval; in the past, the independent variable was allowed to range over all real numbers. See [3], =-=[5]-=-, [9], [10], [15], [17], [27] for details about Dirichlet’s work. 6. “Pathological” Functions. With his new example D�x�, Dirichlet “let the genie escape from the bottle.” A flood of “pathological” fu... |

1 | Infinite Processes - Gardiner - 1982 |

1 |
Lebesgue’s Theory of Integration—Its Origins and Development
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Citation Context ...of function, was among the first to restrict explicitly the domain of the function to an interval; in the past, the independent variable was allowed to range over all real numbers. See [3], [5], [9], =-=[10]-=-, [15], [17], [27] for details about Dirichlet’s work. 6. “Pathological” Functions. With his new example D�x�, Dirichlet “let the genie escape from the bottle.” A flood of “pathological” functions, an... |

1 |
Augustin-Louis Cauchy and the Development of
- Iacobacci
- 1965
(Show Context)
Citation Context ...hatever of this variable and of constants [23, p. 72]. This was the first formal definition of function, although Bernoulli did not explain what “composed in any manner whatever” meant. See [3], [6], =-=[12]-=-, [27] for details of this section. 2. Euler’s Introductio in Analysin Infinitorum. In the first half of the 18th century, we witness a gradual separation of 17th-century analysis from its geometric o... |

1 |
Calculus of the Trigonometric Functions,” Hist
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(Show Context)
Citation Context ... being algebraic or transcendental; single-valued or multivalued; and implicit or explicit. The Introductio contains one of the earliest treatments of trigonometric functions as numerical ratios (see =-=[13]-=-), as well as the earliest algorithmic treatment of logarithms as exponents. The entire approach is algebraic. Not a single picture or drawing appears (in v. 1). Expansions of functions in power serie... |

1 |
Fourier Series: The Genesis and Evolution of a Theory,” The First Herbert Ellsworth Slaught Memorial Paper
- Langer
- 1947
(Show Context)
Citation Context ... equations, seemed to provide a conclusive proof of the fact that functions ‘more general than those expressed by an equation’ were legitimate mathematical objects ... [22, p. 86]. See [3], [4], [9], =-=[16]-=-, [18], [19], [22], [27] for details on section 3. 4. Fourier and Fourier Series. Fourier’s work on heat conduction (submitted to the Paris Academy of Sciences in 1807, but published only in 1822 in h... |

1 |
Vision of a Generalized Partial Differential Calculus for a Generalized
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Citation Context ...ions, seemed to provide a conclusive proof of the fact that functions ‘more general than those expressed by an equation’ were legitimate mathematical objects ... [22, p. 86]. See [3], [4], [9], [16], =-=[18]-=-, [19], [22], [27] for details on section 3. 4. Fourier and Fourier Series. Fourier’s work on heat conduction (submitted to the Paris Academy of Sciences in 1807, but published only in 1822 in his cla... |

1 |
Function” (in Russian), The Great Soviet Encyclopedia
- Luzin
- 1940
(Show Context)
Citation Context ...t the only occasion on which EULER knew examples which did not comply with his conceptions but which he may have considered to be insignificant exceptions from the general rule” [27, p. 67]. See also =-=[19]-=-. 4sIn 1747, d’Alembert solved the Vibrating-String Problem by showing that the motion of the string is governed by the partial differential equation � (a is a constant), the so-called wave equation. ... |

1 |
Vibrating Strings and Arbitrary Functions;” In: The Logic of Personal Knowledge: Essays Presented to M. Polanyi on his Seventieth Birthday, The Free
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- 1961
(Show Context)
Citation Context ... to provide a conclusive proof of the fact that functions ‘more general than those expressed by an equation’ were legitimate mathematical objects ... [22, p. 86]. See [3], [4], [9], [16], [18], [19], =-=[22]-=-, [27] for details on section 3. 4. Fourier and Fourier Series. Fourier’s work on heat conduction (submitted to the Paris Academy of Sciences in 1807, but published only in 1822 in his classic Analyti... |

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1 |
Review of The Analysis of Linear Partial Differential Operators by
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- 1984
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Citation Context ...omes from such a function: The distribution given by ��x� � x�0� corresponds to the “Dirac -function” mentioned above, and does not arise from any function F in the way described above. See[4], [18], =-=[26]-=-. A basic property of distributions is that each distribution has a derivative that is again a 14 distribution. In fact, � �� F�t�x�t� dt. � ��x� �a, b�. F : D → � � 13 The following is a heuristic ar... |