# 5. Continuity of multivariate functions

## Definition: Metric Spaces

A set $$X$$ is said to be endowed with a metric or $$a$$ **metric space structure** or to be **a metric space** if a function

$$
d: X \times X \rightarrow \mathbb{R}
$$

is exhibited satisfying the following conditions:&#x20;

1. $$d\left(x_{1}, x_{2}\right)=0 \Leftrightarrow x_{1}=x_{2}$$
2. $$d\left(x_{1}, x_{2}\right)=d\left(x_{2}, x_{1}\right)$$ (symmetry)&#x20;
3. $$d\left(x_{1}, x_{3}\right) \leq d\left(x_{1}, x_{2}\right)+d\left(x_{2}, x_{3}\right)$$ (the triangle inequality),

where $$x_{1}, x_{2}, x_{3}$$ are arbitrary elements of $$X$$.



**By default, all the Spaces mentioned below are Metric Spaces.**



### traditional distance:

$$
d\left(x_{1}, x_{2}\right)=\sqrt{\sum_{i=1}^{n}\left|x_{1}^{i}-x_{2}^{i}\right|^{2}}
$$

between points $$x_{1}=\left(x_{1}^{1}, \ldots, x_{1}^{n}\right)$$  and $$x_{2}=\left(x_{2}^{1}, \ldots, x_{2}^{n}\right)$$ in $$\mathbb{R}^{n}$$, this can also introduce the distance

$$
d_{p}\left(x_{1}, x_{2}\right)=\left(\sum_{i=1}^{n}\left|x_{1}^{i}-x_{2}^{i}\right|^{p}\right)^{1 / p}
$$

where $$p \geq 1$$.&#x20;

### Minkowski's inequality

$$
\left(\sum_{i=1}^{n}\left|x^{i}+y^{i}\right|^{p}\right)^{\frac{1}{p}} \leq\left(\sum_{i=1}^{n}\left|x^{i}\right|^{p}\right)^{\frac{1}{p}}+\left(\sum_{i=1}^{n}\left|y^{i}\right|^{p}\right)^{\frac{1}{p}}, \quad \forall x, y \in \mathbb{R}^{n}, p \geq 1
$$

so $$d_{p}\left(x_{1}, x_{2}\right)$$ is satisfying the following conditions above.&#x20;

### Definition: neighborhood

> Chinese: 鄰域

For $$\delta>0$$ and $$a \in X$$ the set

$$
B(a, \delta)=\{x \in X \mid d(a, x)<\delta\}
$$

is called the ball with center $$a \in X$$ of radius $$\delta$$ or the $$\delta -$$neighborhood of the point $$a$$.

##

\>>>>>>>>>>>>>>>>>> Under construction <<<<<<<<<<<<<<<<<<

### Accumulation point

> Chinese: 聚點



### Limit point

> Chinese: 極限點



### Boundary point

> Chinese: 邊界點



### Open set

> Chinese: 開集

A subset $$U$$ of a metric space $$(M, d)$$ is called open if, given any point $$x$$ in $$U$$, there exists a real number $$\varepsilon>0$$ such that, given any point $$y$$ in $$M$$ with $$d(x, y)<\varepsilon$$, $$y$$ also belongs to $$U$$.&#x20;

### Derived set

> Chinese: 導集





### Closure

> Chinese: 閉包



### Close Set

> Chinese: 閉集

###

### Connected set

> Chinese: 連通集



### path connectedness

> Chinese: 道路連通



## &#x20;**Quasi**-**mean value theorem**





## The rank theorem

>



$$\blacksquare$$
