### 数学代写|数值方法作业代写numerical methods代考| Uniform Continuity

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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|Uniform Continuity

In general terms, uniform continuity guarantees that $f(x)$ and $f(y)$ can be made as close to each other as we please by requiring that $x$ and $y$ be sufficiently close to each other. This is in contrast to ordinary continuity, where the distance between $f(x)$ and $f(y)$ may depend on $x$ and $y$ themselves. In other words, in Definition $1.1 \delta$ depends only on $\epsilon$ and not on the points in the domain. Continuity itself is a local property because a function $f$ is or is not continuous at a particular point and continuity can be determined by looking at the values of the function in an arbitrary small neighbourhood of that point. Uniform continuity, on the other hand, is a global property of $f$ because the definition

refers to pairs of points rather than individual points. The new definition in this case for a function $f$ defined in an interval $I$ is:
$$\forall \varepsilon>0 \exists \delta>0 \text { s.t. } \forall x, y \in I:|x-y|<\delta \Rightarrow|f(x)-f(y)|<\epsilon .$$ Let us take an example of a uniformly continuous function: $$f: \mathbb{R} \rightarrow \mathbb{R}, \quad f(x)=3 x+7$$ Then $|f(x)-f(y)|=|3 x+7-(3 y+7)|=3|x-y|<3 \delta<\epsilon, \quad(x, y \in \mathbb{R})$. Choose $\delta=\epsilon / 3$. In general, a continuous function on a closed interval is uniformly continuous. An example is: $$f(x)=x^{2} \text { on } I=[0,2]$$ Let $x, y \in I$. Then: $$|f(x)-f(y)|=(x+y)|x-y|<(2+2) \delta=\epsilon .$$ Choose $\delta=\epsilon / 4$. An example of a function that is continuous and nowhere differentiable is the Weierstrass function that we can write as a Fourier series: $$f(x)=\sum_{n=0}^{\infty} a^{n} \cos \left(b^{n} \pi x\right), \quad 01+\frac{3}{2} \pi. This is a jagged function that appears in models of Brownian motion. Each partial sum is continuous, and hence by the uniform limit theorem (which states that the uniform limit of any sequence of continuous functions is continuous), the series (1.6) is continuous. ## 数学代写|数值方法作业代写numerical methods代考|Classes of Discontinuous Functions A function that is not continuous at some point is said to be discontinuous at that point. For example, the Heaviside function (1.2) is not continuous at x=0. In order to determine if a function is continuous at a point x in an interval (a, b) we apply the test:$$
\begin{aligned}
&f(x+)=q \text { if } f\left(t_{n}\right) \rightarrow q, n \rightarrow \infty \text { for all sequences }\left{t_{n}\right} \text { in }(x, b) \text { s.t. } t_{n} \rightarrow x \
&f(x-)=q \text { if } f\left(t_{n}\right) \rightarrow q, n \rightarrow \infty \text { for all sequences }\left{t_{n}\right} \text { in }(a, x) \text { s.t. } t_{n} \rightarrow x \
&\exists \lim {t \rightarrow x} f(t) \Leftrightarrow f(x+)=f(x-)=\lim {t \rightarrow x} f(t)=f(x) .
\end{aligned}
$$There are two (simple discontinuity) main categories of discontinuous functions: • First kind: f(x+)=\lim {t \rightarrow x+} f(t) and f(x-)=\lim {t \rightarrow x-} f(t) exists. Then either we have f(x+) \neq f(x-) or f(x+)=f(x-) \neq f(x). • Second kind: a discontinuity that is not of the first kind. Examples are:$$
\begin{aligned}
&f(x)=\left{\begin{array}{l}
1, x \text { rational }(x \in \mathbb{Q}) \
0, x \text { not rational, } x \notin \mathbb{Q} \
\text { 2nd kind: Neither } f(x+) \text { nor } f(x-) \text { exists. }
\end{array}\right. \
x+2, \quad 0 \leq x<1 \
\text { Simple discontinuity at } x=0 .\end{cases}
\end{aligned}
$$You can check that this latter function has a discontinuity of the first kind at x=0. ## 数学代写|数值方法作业代写numerical methods代考|DIFFERENTIAL CALCULUS The derivative of a function is one of its fundamental properties. It represents the rate of change of the slope of the function: in other words, how fast the function changes with respect to changes in the independent variable. We focus on real-valued functions of a real variable. Let f: \mathbb{R} \rightarrow \mathbb{R}. Then the derivative of f at x (if it exists) is defined by the limit for x \in[a, b] : \varphi(t)=\frac{f(t)-f(x)}{t-x}(t \neq x), f^{\prime}(x)=\lim {t \rightarrow x} \varphi(t) or \frac{d f(x)}{d x}=f^{\prime}(x)=\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h} . This limit may not exist at certain points, and it is possible to define right-hand and left-hand limits, that is, one-sided derivatives. Some results that we learn in high school are:$$
\begin{aligned}
&(f+g)^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x) \
&(f g)^{\prime}(x)=f^{\prime}(x) g(x)+f(x) g^{\prime}(x) \
&\left(\frac{f}{g}\right)^{\prime}(x)=\frac{g(x) f^{\prime}(x)-g^{\prime}(x) f(x)}{g^{2}(x)}(g(x) \neq 0)
\end{aligned}
$$A composite function is a function that we can differentiate using the chain rule that we state as follows: x \in[a, b], \quad \exists f^{\prime}(x) with g differentiable at f(x). Then:$$
\begin{aligned}
&h(t) \equiv g(f(t)), \quad a \leq t \leq b \text { has derivative } \
&h^{\prime}(x)=g^{\prime}(f(x)) f^{\prime}(x) .
\end{aligned}
$$A simple example of use is:$$
\begin{aligned}
f(x) &=x^{2}, \quad g(y)=2 y+1 \
h(x) &=g(f(x))=g\left(x^{2}\right)=2 x^{2}+1 \
h^{\prime}(x) &=g^{\prime}(f(x)) f^{\prime}(x)=4 x(=2 * 2 x)
\end{aligned}
$$More challenging examples of composite functions are:$$
\begin{aligned}
&f(x)=\left{\begin{array}{l}
x \sin \frac{1}{x}, \quad x \neq 0 \
\end{array}\right. \
&f^{\prime}(x)=\sin \frac{1}{x}-\frac{1}{x} \cos \frac{1}{x}, \quad x \neq 0
\end{aligned}

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## MATLAB代写

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