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.

Fifth Chapter.

Geometrical representation of a function; its continuity and its
differential quotient.
15. By the help of the Cartesian system of coordinates - most
simply by a rectangular system - we present to ourselves an image
of the entire course of a function, as a succession of points, whose
number can be arbitrarily increased and whose uninterrupted connexion
can be considered generally as a curve, when we lay off each value
of x as a length along the axis of abscissa, and each corresponding y
as a perpendicular ordinate at the extremity of x. The extremity of
this ordinate is the point which corresponds to the system of values x,
y. But since points even though infinitely numerous never generate a
curve, but it must always be made up of lines between points, as a
matter of fact the only way we obtain an image of the function is,
by constructing arbitrarily many separate points corresponding to different systems of values, and connecting these points by right lines. This
approximate image of the function, a polygon with arbitrarily many
angles, will present a certain general view of the whole course of the
function, which will be more correct the more we increase the number
of points constructed; but in its individual small parts this representation will never be quite exact. Specially, when the function geometrically represented is very much twisted near any point, i. e. when it
presents polygons with salient and reentrant angles starting in and out,
every image we thus procure will exhibit its course very imperfectly,
and will undergo very considerable changes as more points are employed in the construction, so that the true image of the function, in
regard to all its properties at each point, cannot be fixed in this manner.
But when we say of a function: it can be exactly represented at
a point by a curve, we are enunciating a definite property for this
point; for we are assuming, that while the number of angles of
the polygon is arbitrarily increased, the directions of the sides of
the polygon proceeding from this point as an angle converge to fixed
limiting positions. We shall in the present Chapter formulate the
condition for this analytically.