Early attempts to define a function were made by James Gregory (1687), Euler (1748), and, later in the 18th century, by La Croix, Lagrange, and d'Alembert.
All these attempts were intuitive, rough-and-ready affairs and none gained acceptance.
Fourier and Cauchy, both around 1820, offered improved versions;
finally Dirichlet in 1837 identified the essential property of uniqueness": y is a function of x when to each value of x in a given interval there correspond as unique value of y".
This is not quite the end of the story, of course;in time it became apparentt.
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