🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

For a function mapping $S\subseteq {\mathbb{R}}^n$ into ${\mathbb{R}}^m$ the following are equivalent

• $f$ is continuous on $S$
• For every convergent sequence $\left(x_n\right):{\mathbb{N}}_1\to {\mathbb{R}}^n$ such that $\left(x_n\right)\to a\in S$ then $${\displaystyle {lim}_{n\to \infty }f\left(x_n\right)=f\left(a\right)}$$
• For every open set $U\subseteq {\mathbb{R}}^m$, the set $f^{-1}\left(U\right)$ is open in $S$