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