notes/docs/mathematics/multivariable-calculus/limits-and-continuity.md

Limits and continuity

Limit

Definition: let D \subseteq \mathbb{R}^m and let f: D \to \mathbb{R}^n, with m,n \in \mathbb{N}. Let \mathbf{a} be the point \mathbf{x} approaches, then f approaches the limit L \in \mathbb{R}^n

\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L \iff \forall \varepsilon_{>0} \exists \delta_{>0} \Big[0 < |\mathbf{x} - \mathbf{a}|< \delta \implies |f(\mathbf{x}) - L| < \varepsilon \Big],

with \mathbf{a}, \mathbf{x} \in \mathbb{R}^m.

Continuity

Definition: let D \subseteq \mathbb{R}^m and let f: D \to \mathbb{R}^n, with m,n \in \mathbb{N}. Then f is called continuous at \mathbf{a} if

\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a}),

with \mathbf{a}, \mathbf{x} \in \mathbb{R}^m.