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**Johann Bernoulli**(1718)

- I call a function of a variable magnitude a quantity composed in any manner whatsoever from this variable magnitude and from constants.
**Euler**(1748)

- A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
**Euler**(1775)

- When quantities depend on each other in such a way that [the former] undergo changes themselves when [the latter] change, the [the former] are called functions of [the latter]. This is a very comprehensive idea which includes in itself all the ways in which one quantity can be determined by others.
**Lacroix**(1810)

- Every quantity whose value depends on one or more quantities is called a function of these latter, whether one knows or is ignorant of what operations it is necessary to use to arrive from the latter to the first.
**Fourier**(1822)

- In general the function
*f*(*x*) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given to the abscissa*x*, there is an equal number of ordinates*f*(*x*). All have actual numerical values, either positive or negative or null. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as if it were a single quantity. **Heine**(1872)

- A single-valued function of a variable
*x*is an expression which for every single rational or irrational value of*x*is uniquely determined. **Dedekind**(1888)

- A function
*φ*on a set*S*is a law according to which to every determinate element*s*of*S*there belongs a determinate thing which is called the transform of*s*and denoted*φ*(*s*).

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