# 数学代写|勒贝格积分代写Lebesgue Integration代考|MAT00013H

#### Doug I. Jones

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Hankel’s Types of Discontinuity

It is clear that Hankel was very impressed by Riemann’s example of a function (Example 2.1) that is discontinuous at all rational numbers with even denominators, yet is integrable. He sought to understand what happens in general. Using a technique that he dubbed “condensation of singularities,” Hankel showed how to take a function with a singularity at one point, either a discontinuity or an infinite oscillation such as $\sin (1 / x)$ near $x=0$, and use it to construct integrable functions with singularities at every rational number. Cantor would later simplify this method and show how to apply it to any countable set. What is significant for our purposes is that both the set of points at which the function is continuous and the set of points at which the function is discontinuous are dense.

The rational numbers are dense in $\mathbb{R}$. The rational numbers with even denominators are also dense. So are the irrational numbers.

Hankel noticed that all of his examples of integrable functions that are discontinuous on a dense set of points have the property that the set of points of continuity is also dense. The examples that we have seen so far of Riemann integrable functions that are discontinuous on a dense set of points include Riemann’s function (Example 2.1), the function $g$ in Exercise 2.1.5, and the function $m$ in Exercise 2.1.14. What characterizes all of these examples as well as the others that Hankel found is that the set of points of continuity are also dense. This suggested to him that he should separate discontinuous functions into two classes: those for which the points of continuity are not dense and those for which the points of continuity are dense.

Thus, for example, Dirichlet’s function (Example 1.1) is totally discontinuous since it is discontinuous at every point.

All of the examples that we have seen so far of Riemann integrable functions that are discontinuous on a dense set of points are pointwise discontinuous. Hankel believed that every pointwise discontinuous function must be Riemann integrable.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Hankel’s Error

Hankel’s argument for his assertion that every pointwise discontinuous function is Riemann integrable is not unreasonable. We choose an arbitrary $\sigma>0$. If a function is continuous at one value, then we can find an interval around this value on which the oscillation is less than $\sigma$. If the set of points of continuity is dense, then we have succeeded in putting each element of this dense set inside an interval that contains no points of $S_\sigma$.

Those points with oscillation larger than $\sigma$ constitute a very thin set. Between any two points of this set there must be an entire open interval of points not in the set. Hankel believed that such a set must have outer content 0 and therefore must be Riemann integrable. This belief was reinforced by the fact that all of the examples of pointwise discontinuous functions that Hankel knew, examples such as Riemann’s function, were integrable.

Hankel’s fallacy, and he was not the only prominent mathematician to fall into it, was to assume that such a thin set cannot have positive outer content. Thomas Hawkins has presented evidence that between 1870 and 1875 this was the case for Hankel, for Axel Harnack, and for Paul du Bois-Reymond. But in 1878, when Ulisse Dini published his book on the theory of functions of real variables, Fondamenti per la teorica delle funczioni di variabili reali, Dini expressed doubt in the validity of Hankel’s claim. As we shall see in Chapter 4, finding the flaw in Hankel’s reasoning would greatly advance our understanding of the structure of the real numbers, as it also revealed problems with Riemann’s definition of the integral.

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Hankel’s Types of Discontinuity

Hankel 注意到他的所有在稠密点集上不连续的可积函数示例都具有连续点集也是稠密的属性。到目前为止，我们看到的黎曼可积函数在密集点集上不连续的例子包括黎曼函数（例 2.1）、函数g在练习 2.1.5 中，函数m在练习 2.1.14 中。所有这些例子以及汉克尔发现的其他例子的特征是连续点集也很密集。这向他建议，他应该将不连续函数分为两类：连续点不密集的函数和连续点密集的函数。

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Hankel’s Error

Hankel 断言每个点不连续函数都是黎曼可积的论点并非不合理。我们任意选择σ>0. 如果一个函数在一个值上是连续的，那么我们可以找到一个围绕这个值的区间，在该区间上振荡小于σ. 如果连续点集是稠密的，那么我们已经成功地将这个稠密集的每个元素放在一个不包含点的区间内Sσ.

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