02 - Continuity in Higher Dimensions

Continuous Function

Function $f:\Omega \subseteq {\mathbb{R}}^n\to {\mathbb{R}}^m$ is continuous at $\underline{a}\in \Omega$ if

$$\forall ϵ>0,\exists \delta >0\text{ such that }\forall \underline{x}\in \Omega \text{ with }|\underline{x}-\underline{a}|<\delta ⟹|f(\underline{x})-f(\underline{a})|<ϵ$$

Which also means that

$$f(\underline{x})-f(\underline{a})\to \underline{0}\text{ as }\underline{x}\to \underline{a}$$