$\Theta \rho ϵ\eta \Pi \alpha \tau \pi$

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

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