![]() | The Garden of Archimedes
A Museum for Mathematics |
My studies abroad having allowed me to attend the courses on Analysis at the University of Berlin in the year 1877-78, I felt almost obliged to let my study companions at least in part into the knowledge of the new views and concepts that Prof. Weierstrass has been introducing into the sciences, which are becoming widespread in Germany due to the work of his numerous disciples, but tend to be almost unknown to Italian students, due to the well known aversion of that master to print. That which prevented me from attempting to publish was the difficulty in finding a convenient way of expounding such topics so sensitive and subject to controversy for their novelty, that a word used improperly was enough to distort any concept [...]
In the points or the intervals in which there is no derivative of a function, or where one is uncertain at least as to whether it exists, it not being possible to consider together, and at times not even separately, the limits of the ratiofor
tending to zero for positive and negative values, it will be natural to come to examine this relation directly for every value of
between
and
, or at least the limits within which this ratio oscillates as
decreases indefinitely , and this considering separately the one corresponding to positive values of
from the one corresponding to negative values; thus one reaches very general results, some of which include as special cases, also many of those we have already obtained. To be brief, we shall thus call the incremental ratio
; and we shall call right incremental relation the one corresponding to the
positive , whilst the one corresponding to
negative shall be called left incremental ratio [...]
We say that tendingto
,
has limit
, when if a small quantity
is fixed arbitrarily, a quantity
can be determined such that for every value of
, that differs from
less than
, the absolute value of
is less than
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We say that asgrows indefinitely,
has limit
, if when a small quantity
is fixed arbitrarily, a number
can be determined such that for every value of
will be
true in absolute value.[...]