### 数学代写|数值方法作业代写numerical methods代考|Real Analysis Foundations for this Book

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值方法作业代写numerical methods代考|INTRODUCTION AND OBJECTIVES

In this chapter we introduce a number of mathematical concepts and methods that underlie many of the topics in this book. The most urgent attention points revolve around functions of real variables, their properties and the ways they are used in applications. We discuss the most important topics from real analysis to help us in our understanding of partial differential equations (PDEs). A definition of real analysis is:
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.
(Wikipedia)
A related branch of mathematics is calculus, which we learn at school:
Calculus, originally called infinitesimal calculus or ‘the calculus of infinitesimals’, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.
(Wikipedia)
In practice, there is a distinction between calculus and real analysis. Calculus entails techniques (and tricks) to differentiate and integrate functions. It does not discuss the conditions under which a function is continuous or differentiable. It assumes that it is allowed to carry out these operations on functions. Real analysis, on the other hand, does discuss these issues and more; for example:

• Continuous functions: How do we recognise them and prove that a function is continuous?
= The different kinds of discontinuous functions.
• Differential calculus from a real-analysis viewpoint.
• Taylor’s theorem.
= An introduction to metric spaces and Cauchy sequences.
In our opinion, these topics are necessary prerequisites for the rest of this book. Knowledge of vector (linear) analysis and numerical linear algebra is also a prerequisite for computational finance. To this end, we devote Chapters 4 and 5 to these topics. Finally, complex variables and complex functions (which are at the heart of complex analysis) are introduced in Chapter 16 . We use the notation $\forall$ to mean ‘for all’ and $\exists$ to mean ‘there exists’.

## 数学代写|数值方法作业代写numerical methods代考|CONTINUOUS FUNCTIONS

In this section we are mainly concerned with real-valued functions of a real variable, that is $f: \mathbb{R} \rightarrow \mathbb{R}$. In rough terms, a continuous function is one that can be drawn by hand without taking the pen from paper. In other words, a continuous function does not have jumps or breaks, but it is allowed to have sharp bends and kinks. Examples of continuous functions are:
\begin{aligned} &f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=x^{2} \ &f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\max (0, x) \end{aligned}
We can see that these functions are continuous just by drawing them. The first function is ‘smoother’ than the second function, the latter being similar to a one-factor call or put payoff on the one hand and a Rectified Linear Unit (ReLU) activation function

on the other hand (Goodfellow, Bengio and Courville (2016)). Intuitively, a function $f$ is continuous if $f(x) \rightarrow f(p)$ when $x \rightarrow p$, no matter how $x$ approaches $p$. Alternatively, small changes in $x$ lead to small changes in $f(x)$.

If we formally differentiate the above ReLU function (1.1), we get the famous discontinuous Heaviside function:
$$H(x)=\left{\begin{array}{l} 0, x<0 \ 1, x \geq 0 \end{array}\right.$$
A discontinuous function is one that is not continuous. Another discontinuous function is:
$x \in \mathbb{R},[x] \equiv$ largest integer $n$ s.t. $n \leq x \leq n+1 .$
Define $f(x)=[x]$; let $p \in \mathbb{Z}$ (integer).
Then taking left and right limits gives different answers, showing that the function is not continuous.

1. $x<p \Rightarrow f(x)=p-1$
2. $x>p \Rightarrow f(x)=p$
Thus $\lim {x \rightarrow p-} f(x)=p-1, \lim {x \rightarrow p+} f(x)=p$.

## 数学代写|数值方法作业代写numerical methods代考|Formal Definition of Continuity

The following definition is based on the fact that small changes in $x$ lead to small changes in $f(x)$.
Definition $1.1$
\begin{aligned} &\lim {x \rightarrow p} f(x)=A \text { means } \forall \varepsilon>0 \exists \delta>0 \text { s.t. } \ &|f(x)-A|<\varepsilon \text { when } 0<|x-p|<\delta . \end{aligned} Some properties of continuous functions $f(x)$ and $g(x)$ are: \begin{aligned} &\lim {x \rightarrow p}(f(x) \pm g(x))=\lim {x \rightarrow p} f(x) \pm \lim {x \rightarrow p} g(x) \ &\lim {x \rightarrow p}(f(x) g(x))=\lim {x \rightarrow p} f(x) \lim {x \rightarrow p} g(x) \ &\lim {x \rightarrow p} \frac{f(x)}{g(x)}=\frac{\lim {x \rightarrow p} f(x)}{\lim {x \rightarrow p} g(x)}, \quad g(x) \neq 0 . \end{aligned}

It can be a mathematical challenge to prove that a function is continuous using the above ‘epsilon-delta’ approach in Definition 1.1. One approach is to use the well-known technique of splitting the problem into several mutually exclusive cases, solving each case separately and then merging the corresponding partial solutions to form the desired solution. To this end, let us examine the square root function:
$$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}, f(x)=\sqrt{x} .$$
We show that there exists $\delta>0$ such that for $x \geq 0$ :
$$|x-y|<\delta \Rightarrow|\sqrt{x}-\sqrt{y}|<\epsilon \forall y \in \mathbb{R}^{+} .$$ Then: $$\sqrt{x}-\sqrt{y}=\frac{x-y}{\sqrt{x}+\sqrt{y}}$$ We now consider two cases: Case $1: x>0$. Then:
$$|x-y|<\delta \Rightarrow|\sqrt{x}-\sqrt{y}| \leq \frac{|x-y|}{\sqrt{x}}=\frac{\delta}{\sqrt{x}}=\epsilon$$
Choose $\delta=\epsilon \sqrt{x}$.
Case $2: x=0$. Then:
$$|x-y|<\delta \Rightarrow|\sqrt{x}-\sqrt{y}|=\frac{|x-y|}{\sqrt{x}+\sqrt{y}}=\frac{y}{\sqrt{y}}=\sqrt{y}=\epsilon$$
Hence:
$$|-y|=|y|<\delta \Rightarrow \sqrt{y}=\epsilon \Rightarrow \sqrt{\delta}<\epsilon \Rightarrow \delta<\epsilon^{2}$$
Choose $\delta=\epsilon^{2}$.
We have thus proved that the square root function is continuous.

## 数学代写|数值方法作业代写numerical methods代考|INTRODUCTION AND OBJECTIVES

（维基百科）

（维基百科）

• 连续函数：我们如何识别它们并证明函数是连续的？
= 不同种类的不连续函数。
• 从实分析的观点看微分。
• 泰勒定理。
= 度量空间和柯西序列的介绍。
我们认为，这些主题是本书其余部分的必要先决条件。矢量（线性）分析和数值线性代数的知识也是计算金融的先决条件。为此，我们将第 4 章和第 5 章专门讨论这些主题。最后，第 16 章介绍了复变量和复函数（它们是复分析的核心）。我们使用符号∀意思是“为所有人”和∃意思是“存在”。

## 数学代写|数值方法作业代写numerical methods代考|CONTINUOUS FUNCTIONS

F:R→R,F(X)=X2 F:R→R,F(X)=最大限度(0,X)

$$H(x)=\left{0,X<0 1,X≥0\对。$$

X∈R,[X]≡最大整数n英石n≤X≤n+1.

1. X<p⇒F(X)=p−1
2. X>p⇒F(X)=p
因此林X→p−F(X)=p−1,林X→p+F(X)=p.

## 数学代写|数值方法作业代写numerical methods代考|Formal Definition of Continuity

F:R+→R+,F(X)=X.

|X−是|<d⇒|X−是|<ε∀是∈R+.然后：X−是=X−是X+是我们现在考虑两种情况：1:X>0. 然后：
|X−是|<d⇒|X−是|≤|X−是|X=dX=ε

|X−是|<d⇒|X−是|=|X−是|X+是=是是=是=ε

|−是|=|是|<d⇒是=ε⇒d<ε⇒d<ε2

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## MATLAB代写

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