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 $U$.

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

Quasi-mean value theorem

The rank theorem

$\blacksquare$