Continuous Vectorial Function

Topics: Continuous Function - Vectorial Function

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(definition)

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a vectorial function. We say that $f$ is continuous in $\overline{x_0}$ if $\forall \epsilon >0$, $\exists \delta >0$ such that:

$$f(B_{\delta }(x_0))\subseteq B_{\epsilon }(f(x_0))$$

…(see open ball), or alternatively:

$$\Vert f(\overline{x_0})-f(\overline{x})\Vert <\epsilon$$

…if $\Vert x_0-x\Vert <\delta$.

Continuity in all ${\mathbb{R}}^n$ §

(theorem)

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$. $f$ is continuous in all ${\mathbb{R}}^n$ if and only if $f^{-1}(u)$ is open for all open sets $u$ in ${\mathbb{R}}^m$.

Tip $f^{-1}(u)$

$f^{-1}(u)=\{x\in {\mathbb{R}}^n|f(x)\in u\}$.

Composition §

(corollary, from this theorem)

If $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ and $g:{\mathbb{R}}^k\to {\mathbb{R}}^n$ are continuous, then $f\circ g:{\mathbb{R}}^k\to {\mathbb{R}}^m$ is also continuous.