By William Dunham
Translators: Li Bomin, Wang Jun, Zhang Huaiyong
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Among the parallel mathematicians who emerged in Italy in the second half of the 19th century, Vito Volterra was a well-known figure. Like his compatriots Giopeth Peiano (1858-1932), Jugenio Bertrani (1835-1900) and Julius Dini (1845-1918), he left an indelible mark on the history of science for his contributions to applied sciences such as electrostatics and fluid mechanics, as well as to theoretical studies such as mathematical analysis. Naturally, what we are looking at here is his latter contribution.
Although Volterra was born on the shores of the Adriatic Sea, he grew up in Florence, central Italy, the birthplace of the Italian Renaissance movement. In Florence, Volterra wandered the streets where the great artist Michelangelo strolled, attending a school named after the poet Dante and the scientist Galileo Galilei. The atmosphere of the Florentine Renaissance in the 15th and 16th centuries seemed to seep into his marrow, because Volterra loved art, literature and music as much as he loved science. He seemed to be a learned Renaissance figure, although three centuries had passed since the movement that began in Florence.
In addition to these causes, his political courage is also commendable. Volterra, who witnessed the rise and rise to power of fascist leader Mussolini in the 1920s, took a stand of open opposition and signed a statement of resistance to the regime. This move eventually cost him his job, but it made him a hero in the intellectual circles of Italy's time. Until his death in 1940, Italy was not free from the scourge of fascist rule, but Volterra had fought valiantly for the expected bright future.
If Volterra showed great courage in his later years, he showed great wisdom in his youth. As early as 11 years old, Volterra had read university-level mathematics textbooks, impressing his adolescent teachers and, in high school, had somehow sought a position as an experimental physics assistant at the University of Florence. His academic development took precedence and peaked in his doctoral dissertation in physics at the age of 22.
In this chapter we discuss two of Volterra's early discoveries, both published in 1881, three years after he graduated from high school. The first discovery was to find a sick function, add a new instance to the ever-increasing list of counterexamples of pathological functions, and find a hole in the Riemann integral that had never been noticed before. The second finding, which seems contradictory but is actually correct, is to obtain a proof that the sick function has its own limits, because Volterra proved the theorem that there can be no function that is continuous at every rational point and discontinuous at every irrational point. Such a function cannot exist due to excessive pathology. We will examine this theorem in its entirety, but we shall begin with a brief description of the counterexample of the pathological function.
Volterra's sick function
We have seen the second form of the fundamental theorem of calculus in chapter 6, as stated by Cauchy as follows:
Not strictly speaking, this indicates that the integral of the derivative under the correct conditions returns to its original function. Cauchy used two assumptions in his proof of theorem : ( a ) that the function F has a derivative , and ( b ) that the derivative itself is continuous. But are these two assumptions necessary?
Hypothesis (a) seems indispensable, since we cannot expect to integrate a derivative in the absence of a derivative. However, the necessity of hypothesis (b) is questionable. In order for the fundamental theorem to hold, do we have to make such strict limits as the continuity of F??
This is not an insignificant issue. As we have seen in chapter 10, the continuity of derivatives cannot be taken for granted, because of functions
There is a discontinuous derivative. On the other hand , we don't need to use the continuity of derivatives as a guarantee of the existence of an integral , because it is easy to find functions whose derivatives are discontinuous but are integrable.
The problem, then, is that if there are any conditions, then what conditions should we add to the derivative F' to guarantee that the fundamental theorem holds. Past discoveries have provided mathematicians with a perspective on the problem that Cauchy did not have, so it seems worth revisiting this important theorem.
This is already a step forward, but there is still a question about F', that is, whether we need to make any other assumptions about it that differ from existence. Perhaps derivatives are intrinsic to the integrable progeny of their specific properties. If so, we can discard both assumptions (b) and (b') and build the fundamental theorems of calculus on the basis of assumptions (a) alone. That would be a less restrictive and more elegant situation.
This is specialized into the question: What undesirable properties might the derivative be allowed? In chapter 10 we proved Dabb's theorem, where the derivative is not required to be continuous, it must have a mediator property. At that point, the derivative appears to be completely "compliant," and mathematicians may speculate that such conformity will include integrability.
We will not examine his example of a sick function in detail, partly because the function is complex, and partly because it may be sufficient to discuss only one sick function (the weierstrass sick function) in one chapter.
How can mathematicians react to Volterra's peculiar example? One option is to accept this result and move on. When applying this theorem, simply append an additional hypothesis about the derivative F'. This will be the path with the least resistance.
However, there is another option. As we have seen before, Riemann points are not guaranteed
Today, Volterra has dashed any hope of establishing a simple calculus fundamental theorem. Toward the end of the 19th century, there were more reasons than ever to suspect that trouble was hidden in the definition of the Riemann integral rather than in the inherent nature of analytics. A few bold people, in order to save the above fundamental theorem, were pushed by Volterra's pathological function to give up the Riemann integral. But I won't continue the discussion.
Hankel's function classification
By the 1880s, analytics had been greatly impacted by pathological function counterexamples that looked more peculiar than the last. The counterexamples we've seen include the following three.
This scenario shows the chaos that exists in analytics and evokes order and order for such a disorganized mathematical status quo.
The person who struggled to accomplish this task was Herman Hankel (1839-1873). He was an admirer of Riemann, who believed that functions should be classified in some similar way, as biologists or geologists did. Hankel proposed such a classification in 1870 (three years before his early death). He hopes to use this classification to elucidate the nature and limits of mathematical analysis.
Hanker examines the families of functions consisting of all bounded functions defined on an interval [a , b] and distinguishes them by their continuity and discontinuous properties. To understand how he categorizes it, we recall a familiar definition that Georg Cantor once gave.
Definition For a set of real numbers A , if at least one element of A is included in any open interval, A is said to be dense.
Two basic examples of dense densities are sets of rational and irrational numbers, since any open interval contains an infinite number of rational and irrational numbers. The name dense is revelatory, indicating that its elements are so closely clustered that they are always contiguous.
With this preparation, we can classify the function as Hankel. He listed functions with all consecutive points on the interval [a, b] as class 1 functions. These functions have good characteristics, they reach maximum and minimum values on intervals, have medial characteristics, and are able to integrate. In Hankel's classification, class 1 functions are at the top of the "food chain."
His class 2 functions include those that are continuous except for a finite number of points on [a, b]. Such functions carry more uncertainty, but their singularity is quantitatively limited and largely under control. An example is the function we saw in chapter 10 defined on the interval [-1, 1].
Because it has a discontinuous point at x =0. Alternatively, we can take a continuous function defined on the interval [a , b] and then redefine it as discontinuous at 50 points, for example, introducing 50 discontinuities. Such a function would belong to Hankel's class 2 functions.
Logically, there is only one class of functions left, namely those with infinite discontinuous points on [a , b]. Naturally, these functions are the worst, but Hankel believes that they can be reclassified into two categories: poor and very bad:
Class 3A: Functions that are not continuous at infinite points in [a , b] but are still continuous in one of the dense densities. He called such functions "functions of point-state discontinuities."
Class 3B: All other discontinuous functions. Hankel calls it a "completely discontiguous function."
We see that a point discontinuous function in a class 3A function, despite the existence of an infinite number of discontinuities, is still continuous in some parts of any open interval. On the other hand, for a function in a class 3B, there must be some open sub-interval (c, d) within the interval (a , b ), in which the function has no continuous points at all. Thus, a completely discontinuous function is characterized by the presence of only discontinuous points on an uninterrupted subevalle.
I would like to ask which of the three pathological functions introduced earlier belong to the Hankel classification scheme? Dirichlet functions are functions that are discontinuous everywhere and belong to the completely discontinuous class 3B functions. Ruler functions are discontinuous at infinite points (rational points), whereas at a dense density (irrational points) they are continuous, and therefore belong to the AAA class of functions with point discontinuities. The Weierstrass function is perhaps the most peculiar, and its classification into class 1 functions seems inappropriate but is actually correct because it is continuous everywhere. Hankel found that his function classification was important in the sense that he knew that class 1 and class 2 functions were Riemann integrable, and that the examples of point-state discontinuous functions that he knew best were also integrable. Conversely , the completely discontinuous Dirichlet function is non-accumulative. For him, the gulf between class 3A functions and class 3B functions seems insurmountable. As Thomas Hawkins points out, "By determining the difference between point-state discontinuous functions and completely discontinuous functions, Hanker believes that he has separated the functions that can be handled by mathematical analysis from those that it is unable to handle."
To show the full value of doing so, Hanker proved a striking theorem that a bounded function on the interval [a , b] is Riemann's integrable, if and only if it is not worse than a point-state discontinuous function. This means that a bounded function can be integral as long as it belongs to class 1, class 2, or class 3A functions; Functions that belong to the 3B class are not integral and cannot be resolved.
Hanker's theorem seems to answer the main question we asked earlier: to what extent can an integrable function be discontinuous? According to him, the answer is "in the worst case, the dot state is discontinuous.". His proof shows that as long as a function is continuous at a dense density, all those points that are not continuous are insignificant for integrity. This is exactly the concise result that mathematicians dream of.
Unfortunately, this is also incorrect.
In this intricate concept, even university inquirers inevitably make mistakes, but Hankel made a prominent mistake. To be fair, half of his theorem is true: if a function is Riemann's productable, then it must indeed be continuous under a dense congruence. A completely discontinuous function has an uninterrupted sub-interval consisting of discontinuous points , and it is impossible to have a Riemann integral. On this point, the Dirichlet function is again thought of.
However, Hanker's proof of the opposite conclusion is flawed. The English mathematician Henry John Stephen Smith (1826-1883) published an example of a function with a point state discontinuity, which is not cumulative. He points out: "This function is noteworthy because it is opposed to the theory of discontinuous functions, which was acquired by the remarkable geometrist Hermann Schwartz. Dr. Hankel's approval that his recent untimely death was a great loss to the mathematical sciences. Smith's example is extraordinary, involving what we now call a nowhere dense structure with positive measurements. For readers who need details, see Thomas Hawkins' book. For the time being, we will confine ourselves to pointing out that the relationship between the continuity of functions and the integrability of Riemann is still unclear, and still does not solve the question of the extent to which an integrable function may be discontinuous. Whatever the value of point state discontinuities, it does not provide an answer to the connection that people have been looking for for a long time.
However, some progress has been made. Riemann had extended the concept of integrability to certain highly discontinuous functions, and the correct half of Hankel's theorem and Smith's function counterexample proved that Riemann's integrability functions were completely contained within a larger set of functions, i.e., a set of continuous functions on a dense density. Incidentally, we note that the term "point state discontinuity" is sometimes used carelessly to mean "in the worst case a point state discontinuity". That is to say, all functions belonging to Hankel's class 1, 2, or A are grouped under the single heading of point-state discontinuities, which leads to the bizarre situation of placing continuous functions (class 1 functions) in the category of "point-state discontinuities". Since a common feature of the previous three types of functions is that they are continuous in dense density, we can think of densely continuous as a term that encapsulates all the functions in classes 1, 2, and AAA.
In any case, at first glance, Hankel's function classification seems like a promising tool for separating functions that analytics can handle from functions that are difficult for analytics to deal with. The result, however, is that many intractable functions are handled very cleverly in the range of set theory and Lebeg integral. Today, Hankel's functional classification is mostly shelved. But in the late 19th century, point state discontinuities remained the object of study by the best mathematicians. Vito Volterra, 21, is one of these mathematicians.
Limits of the sick function
The prevalence of pathological functions shows that any property of a function, no matter how bizarre it may be, can be recognized through examples carefully constructed by a highly creative mathematician.
For example, who would have imagined that a ruler function would be continuous at every irrational point and discontinuous at every rational point? Furthermore, why not assume that somewhere there is an equally bizarre function to be discovered, which is continuous at each rational point and not at each irrational point? It doesn't seem like one example is weirder than another.
From the following two examples, it is clear that the continuity points of a function are sometimes interchangeable with the discontinuities. Start by defining the function
Volterra devoted himself to solving this problem in his 1881 paper " Notes on The Discontinuous Function of Point States " . The result is a strong theorem and two important inferences.
Theorem it is impossible to have two point-state discontinuities on the interval (a , b ), where the continuity point of one function is the discontinuity point of the other function, and vice versa.
The explicit requirement is closed and interval. This is an oversight that can be easily remedied, as we did above. Second, in the previous example, the continuity point of function H is the discontinuity point of function K, and vice versa, we notice that K is a completely discontinuous function (Hankel's class 3B function) rather than a point-state discontinuous function (Hankel's class 3A function). Therefore, that example does not contradict the results of Volterra's proof, which I am afraid will make it difficult for anyone to sleep.
Volterra draws two important inferences from his theorem. The first inference solves a major problem in analytics, which we state below.
Corollary 1 Since there is a function that is continuous at each irrational point and discontinuous at each rational number, it is impossible to find a function that is discontinuous at each irrational point and continuous at each rational number.
To flesh out the details of his argument, we imagine a function G(x) whose CG is a rational set of numbers (densely packed). G, then, is a dot state discontinuous. But we have already encountered the extended ruler function R(x), which is also a point discontinuity, and whose CR is an irrational set. Thus the continuity point of G will be the discontinuity point of R, contradicting Volterra's theorem. Therefore, it is impossible for both functions to exist at the same time. Since the ruler function does exist, we have to conclude that function G does not exist. To paraphrase a comment about cowboys in Western films, Volterra's theorem proves that "the city is not big enough to accommodate both of them at the same time" It is logically impossible for a function to be continuous only at rational number points.
Therefore, the pathological function has its limits and is not all-encompassing. No matter how shrewd the mathematicians may be, certain functions remain out of the way, and this is a fact that Volterra confirmed with this ingenious argument. However, he also got an implicit inference that it is impossible to have a continuous function that takes a rational number at an irrational point and in turn takes an irrational number on the rational point.
Corollary 2 does not exist such a continuous function g (x) defined on the set of real numbers , g ( x ) is a rational number when x takes an irrational number , and g ( x ) is an irrational number when x takes a rational number.
Combining these two assertions proves that the function G is continuous at the rational points and discontinuous at the irrational points – a situation that Volterra has just proved impossible! Inference from this that there can be no function like g. Therefore , no continuous function can transform a rational number into an irrational number, and at the same time transform the irrational number into a rational number.
These results remind us in particular that although both rational and irrational sets are densely packed with real numbers, they are inherently interchangeable. As we have seen, Cantor once specifically pointed out the fact that rational numbers are countable and irrational numbers are uncountable, but this is not the case, and mathematicians have found some more subtle differences between the two number systems. One of these differences is the concept of a set of "types". The type of set was a concept proposed by Volterra's gifted student René Bell, the mathematician we will introduce in the next chapter.
With these accounts, we would like to bid farewell to Vito Volterra, who was only 21 years old at the time. In front of him was a far-reaching and brilliant career, and people would see him continue to achieve mathematical success and gain international recognition, and King George V of England even awarded him the title of Honorary Sir. Tracing the second half of Volterra's life, we know that he gave the 19th century the characteristics of a "century of function theory". From Euler's initial ideas of functions, the concept of functions played a major role in the study of Cauchy, Riemann, and Weierstrass, and was then passed on to a new generation of Cantor, Hankel, and Volterra himself. Functions are supremely dominant in the science of analysis, and the unexpected possibilities found in functions have surprised mathematicians again and again. As we know, Volterra's two different and fascinating discoveries in 1881 gave him an important place in the stories we narrate. For this young man, 1881 was an extraordinary year.