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I am interested in the history of functions. Why did Euler introduce them? When and why did they become central to mathematics? I know the second question has something to do with the famous Fourier-Cauchy dispute over the heat equation, but don't have any sense of the details.

By the way, I'm asking because I would like to give a short, historically informed explanation to high school students why functions are important.

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Functions occur naturally everywhere. Euler did not introduce them, really. They were around a lot longer than that in various forms. It's true that after Euler and others, the nature of how we treated functions changed a lot though as analysis became a more rigorous subject - and hence so did our notions of what a function really is and what properties it has. –  Cameron Williams Jul 18 '13 at 22:12
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And poor Newton, trying to write Principia twenty-something years before Euler was born, and so never introduced to functions ... –  Peter Smith Jul 18 '13 at 22:16
    
I think the book A Radical Approach to Real Analysis by David M. Bressoud touches on this. –  lhf Jul 18 '13 at 22:58

2 Answers 2

You might find these two articles useful:

http://links.jstor.org/sici?sici=0002-9890%28199801%29105%3A1%3C59%3AFPI%3E2.0.CO%3B2-X

http://links.jstor.org/sici?sici=0002-9890%28199803%29105%3A3%3C263%3AFPI%3E2.0.CO%3B2-9

They are parts I and II of Abe Shenitzer's The Evolution of Function.

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The following is an extract from An Introduction to the History of Mathematics - Howard Eves.

The word $\it{function}$ appears to have been introduced by Leibniz in 1694, originally to express any quantity associated to a curve.

Around 1718, Johann Bernoulli had considered a function as any expression made ​​up of a variable and some constants. Shortly thereafter, Euler considered a function as an equation or formula involving any variables and constants.

The concept of Euler remained unchanged until Joseph Fourier was led to consider, in his research on the propagation of heat, the called trigonometric series. These series involve a form of more general relationship between the variables that had been studied previously. In an attempt to give a function definition broad enough to encompass the form of this relationship, Dirichlet came to the following formulation: a variable is a symbol that represents any element of a set of numbers, if two variables $ x $ and $ $ y are linked in such a way that when a value is assigned to $ x $, automatically matches, for some rule or law, a value to $ y $, then we say that $ y $ is a function of $ x $ . The variable $ x $ is called the independent variable, and the variable $ y $ is called the dependent variable. The possible values ​​that $ x $ can take constitute the domain of the function, and the values assumed ​​for $ y $ constitute the image of the function.

Set theory expanded the concept of function in order to cover relations between any two sets of elements, these elements could be numbers or anything else.

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