After reviewing the often contradictory assertions of other investigators, the author seeks to clarify the evolution of the idea of a function by examining mathematics of Antiquity, the Middle Ages and the Modern Period up to the 19th century. He argues that the Babylonians and the Greeks had no general notion of function, despite their compilation of various tables and use of specific correspondence in science and geometry. Oresme and the Merton Calculators appear to have been the first to make a conscious effort to analyze the variability of one quantity with respect to another. Their geometric and kinematic insights had a significant effect on later work. However, with the increasing application of mathematics to science, during the 16th century, the description of functional dependence by formulae and equations became prominent. This approach was sharpened by Descartes in ‘La geometrie’, and became exceedingly powerful as the ability grew to derive power series expansions. Although the word ‘function’was coined by Leibniz in 1673 with a restricted geometric meaning, by the time of John Bernoulli, a function was considered to be some analytic expression in the variable concerned. While Euler customarily maintained this outlook, his research on the vibrating string and the attendant dispute with d’Alembert, as well as his discovery that sometimes a functional correspondence might be mediated in different ways, led him to distinguish between analytic and nonanalytic functions, and, in 1755, to formulate quite a general definition of function. Euler’s impact on his successors is discussed in considerable detail. In the 19th century, it was left to sort out how valid Euler’s classification was, whether continuity should be an intrinsic part of the definition of function, how pathological functions could be, exactly what functions are capable of some kind of analytic representation, and, in a given context, precisely what restrictions on the class of functions is appropriate. (ZDM/Mathdi)