Continuity of multivariate functions
Continuity of multivariate functions
Let $E$ be a set in the space $\mathbb{R}^m$ and the function $f:E \rightarrow \mathbb{R}^n$ is defined on $E$ and takes values in $\mathbb{R}^n$.
Definition 1:
Given the function $f:E\rightarrow \mathbb{R}^n$. If for any neighborhood $V(f(a))$ of the value $f(a)$ of this function at the point $a$, there always exists a neighborhood $U_{E}(a)$ of the point $a$ in the set $E$ such that its image $f(U_{E}(a))$ lies in $V(f(a))$. We call the function $f:E\rightarrow \mathbb{R}^n$ $\textbf{continuous}$ at $a \in E$.
Hence,
($ f:E\rightarrow \mathbb{R}^n$ is continuous at $a \in E$)$ :=$ $(\forall V(f(a))\exists U_E(a)(f(U_E(a))\subset V(f(a))))$
In fact, the Definition $1$ is identical to our definition of continuity of the real-valued functions. Therefore, we can also give the following expression of this definition as we did there: ($ f:E\rightarrow \mathbb{R}^n$ is continuous at $a \in E$)
$\qquad \qquad$ $:=$ $(\forall \varepsilon > 0 \exists \delta>0\forall x \in E(d(x,a)<\delta \Rightarrow d(f(x),f(a))<\varepsilon))$,
or, if the point $a$ is the limit point of $E$, then
($ f:E\rightarrow \mathbb{R}^n$ is continuous at $a \in E$)$ :=$ ($\lim_{x \in E \to a}f(x)=f(a)$).
Obviously, the concept of continuity makes sense only when the point $a \in E$ is the limit point of the set $E$ and the function $f$ has been defined on $E$.
From Definition 1, we can know that the sufficient necessary condition for $f:E \rightarrow \mathbb{R}^n$(the mapping given by the relation below)
\[(x^1,\cdots,x^m)=x \xmapsto{f} y =(y^1,\cdots,y^n)=(f^1(x^1,\cdots,x^m),\cdots,f^n(x^1,\cdots,x^m))\]to be continuous at a point is that every function $y^i=f^i(x^1,\cdots,x^m)$ is continuous at this point.
Specially, we still remember that the $\textbf{path}$ in $\mathbb{R}^n$ refers to the map $f:I \rightarrow \mathbb{R}$ in $I \subset \mathbb{R}$. It was decribed by the continuous function $f^1(x),\cdots, f^n(x)$:
\[x \mapsto y = (y^1,\cdots,y^n) = (f^1(x),\cdots,f^n(x)).\]Hence, we define that the $\textbf{path}$ in $\mathbb{R}$ is the continuous mapping from the interval $I \subset \mathbb{R}$ on real axis to $\mathbb{R}^n$.
Similar to the definition of the amplitude of a real-valued function at a point, the concept of the amplitude of a function whose value belongs to $\mathbb{R}^n$ at a point can be introduced.
Definition 2:
\(w(f;a):= \lim_{r \to +0} w(f;B_E(a;r))\) is called the $\textbf{amplitude}$ of the function $f:E \rightarrow \mathbb{R}^n$ at $a\in E$.
From the definition of the continuous function from Definition $1$ and the properties of the limit and the Cauchy principal, we obtain a series of common local properties of the continuous function, which are listed below:
a)