“ The general notions of functionality, limits and continuity are explained in the article Function. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.
33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x = t. If certain conditions are satisfied the aggregate of the points determined by the functional relations form a curve. ”

“ When the second differential coefficient exists, or the first is differentiable, we have the relation
The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′ (x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ (x) must be a linear function of x in the interval, is important. ”

“ The function
has been called Bernoulli’s function of the mth order by J. L. Raabe (Crelle’s J. f. Math. Bd. xlii., 1851) . Bernoulli’s numbers and functions are of especial importance in the calculus of finite differences (see the article by D. Seliwanoff in Ency. d. math. Wiss. Bd. i., E., 1901) .
When x is given in terms of y by means of a power series of the form
there arises the problem of expressing y as a power series in x. ”

“ This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated series, and, when the remainder tends to zero as n Taylor’s Theorem. increases, it may be transformed into an infinite series. ”

“ In the last resort the definitions of many Areas. quantities of geometrical import must be analytical, not geometrical. As illustrations of this statement we may take the definitions of the areas and lengths of curves. ”