The Garden of Archimedes  A Museum for Mathematics

Weierstrass and treatises on Analysis in Italy

|    From Pincherle's Essay    |    Dini's derivatives    |    Genocchi's limit    |   

From Pincherle's Essay .

* Exhibit VII. 1

Salvatore Pincherle
Essay of an introduction to the theory of analytic functions according to the principles of Prof. C. Weierstrass

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 [...]

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The derivatives of Dini.

When the Foundations of the theory of functions of a real variable was ready and just about to go for printing, Dini added another paragraph in which he introduced new remarks on the derivatives of a function. In this paragraph he introduces the concept of right and left incremental ratio and relative upper and lower limits, defining the famous "derivatives of Dini".

* Exhibit VII.2

Ulisse Dini
Foundations of the theory of functions of a real variable

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 ratio $\frac{f(x+h)-f(x)}{h}$ for $h$ tending to zero for positive and negative values, it will be natural to come to examine this relation directly for every value of $x$ between $a$ and $b$, or at least the limits within which this ratio oscillates as $h$ decreases indefinitely , and this considering separately the one corresponding to positive values of $h$ 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 $\frac{f(x+h)-f(x)}{h}$; and we shall call right incremental relation the one corresponding to the $h$ positive , whilst the one corresponding to $h$ negative shall be called left incremental ratio [...]

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Genocchi's definition of limit

* Exhibitions VII.3

Angelo Genocchi
Differential calculus and fundamentals of integral calculus published with additions by Dr Giuseppe Peano

We say that tending $x$ to $a$, $y=f(x)$ has limit $A$, when if a small quantity $\varepsilon$ is fixed arbitrarily, a quantity $h$ can be determined such that for every value of $x$, that differs from $a$ less than $h$, the absolute value of $f(x)-A$ is less than $\varepsilon$
We say that as $x$ grows indefinitely, $y=f(x)$ has limit $A$, if when a small quantity $\varepsilon$ is fixed arbitrarily, a number $N$ can be determined such that for every value of $x>N$ will be $f(x)-A<\varepsilon$ true in absolute value.[...]

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