1 Limits and Continuity of Multivariate Functions 1.1 Examples of Multivariate Functions
Multivariate Function
An m m m f : D → R f : D \to \mathbb{R} f : D → R D D D R m \mathbb{R}^m R m y y y x 1 , x 2 , … , x m x_1, x_2, \dots, x_m x 1 , x 2 , … , x m y = f ( x 1 , x 2 , … , x m ) y = f(x_1, x_2, \dots, x_m) y = f ( x 1 , x 2 , … , x m )
Multivariate Mapping
The values of multiple variables y 1 , … , y n y_1, \dots, y_n y 1 , … , y n x 1 , … , x m x_1, \dots, x_m x 1 , … , x m
( y 1 ⋮ y n ) = ( f 1 ( x 1 , … , x m ) ⋮ f n ( x 1 , … , x m ) ) , i.e., { y 1 = f 1 ( x 1 , … , x m ) , ⋮ y n = f n ( x 1 , … , x m ) . \begin{equation*}
\begin{pmatrix}
y_1 \\
\vdots \\
y_n
\end{pmatrix}
=
\begin{pmatrix}
f_1(x_1, \dots, x_m) \\
\vdots \\
f_n(x_1, \dots, x_m)
\end{pmatrix},
\quad \text{i.e.,} \quad
\begin{cases}
y_1 = f_1(x_1, \dots, x_m), \\
\vdots \\
y_n = f_n(x_1, \dots, x_m).
\end{cases}
\end{equation*} y 1 ⋮ y n = f 1 ( x 1 , … , x m ) ⋮ f n ( x 1 , … , x m ) , i.e., ⎩ ⎨ ⎧ y 1 = f 1 ( x 1 , … , x m ) , ⋮ y n = f n ( x 1 , … , x m ) . 1.2 Distance and Limits of Sequences in R m \mathbb{R}^m R m
Definition 1.2.1. A bivariate function d : R m × R m → R d : \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R} d : R m × R m → R R m \mathbb{R}^m R m
For any x , y ∈ R m \mathbf{x}, \mathbf{y} \in \mathbb{R}^m x , y ∈ R m d ( x , y ) = d ( y , x ) ≥ 0 d(\mathbf{x}, \mathbf{y}) = d(\mathbf{y}, \mathbf{x}) \ge 0 d ( x , y ) = d ( y , x ) ≥ 0
For any x , y ∈ R m \mathbf{x}, \mathbf{y} \in \mathbb{R}^m x , y ∈ R m d ( x , y ) = 0 d(\mathbf{x}, \mathbf{y}) = 0 d ( x , y ) = 0 x = y \mathbf{x} = \mathbf{y} x = y
For any x , y , z ∈ R m \mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^m x , y , z ∈ R m d ( x , z ) ≤ d ( x , y ) + d ( y , z ) d(\mathbf{x}, \mathbf{z}) \le d(\mathbf{x}, \mathbf{y}) + d(\mathbf{y}, \mathbf{z}) d ( x , z ) ≤ d ( x , y ) + d ( y , z )
For r > 0 r > 0 r > 0 B ( x , r ) : = { y ∈ R m ∣ d ( x , y ) < r } , B(\mathbf{x}, r) := \{\mathbf{y} \in \mathbb{R}^m \mid d(\mathbf{x}, \mathbf{y}) < r\}, B ( x , r ) := { y ∈ R m ∣ d ( x , y ) < r } , x \mathbf{x} x r r r
A distance d d d R m \mathbb{R}^m R m x , y , z ∈ R m \mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^m x , y , z ∈ R m d ( x + z , y + z ) = d ( x , y ) . d(\mathbf{x} + \mathbf{z}, \mathbf{y} + \mathbf{z}) = d(\mathbf{x}, \mathbf{y}). d ( x + z , y + z ) = d ( x , y ) .