An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

Fourth Chapter.

Conception and notation of functions of a variable.
11. If the value of a variable y is determined by the value of a
variable x in such a way that to each value of x within a certain
interval one or more values of y can be calculated or assigned in any
prescribed manner, then y is called a Function*) of the continuous
variable x within this interval; y is also called the dependent and x
the independent variable, or the argument of the function. If there
be one value of the function for each value of the variable it is called
a single-valued (one-valued or unique) function, if more values
a many-valued (or ambiguous) function.
This dependence may be perfectly casual. the independent variable
serving merely to indicate the position in a Table, the function being
that which is found there: in the problems which deal with measurable
quantities in Physics, the functions with which we are always concerned
are those in which the dependence of the variables is ascertained by
Observation. The dependence may be expressed by an equation between
x and y, which may or may not have for all values of x a sense already defined. For instance, by the relation
y2 (x - 1) (3 - x) or y - + ](x- 1) (3-x)
a two-valued function of x is defined by reason of the double sign, but
it can be calculated in real numbers only for the values, I < x 3.
When the relation between x and y is given in the form of the
general equation f(x, y) = 0 the function is called implicit: but
when the equation is in a form solved for y, y = f(x), we call y
an explicit function of x. The functional connexion between x and
y may also arise by means of a third variable and the function be expressed by a parameter: x= f (t), y (p(t); to each value of t
belongs a value of x and of y; in this way values of x and y become
connected.
It is usual to divide functions into algebraic and transcendental. When the equation by which y is defined can be brought to
*) The term "function" was introduced by John Bernoulli (1667-1748)
(opera omnia t. II p. 241).
2*