# 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: 

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) 
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 $$p \geq 1$$. 

### 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. 

### 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

>

$$\blacksquare$$