Continuity (Real Vector Functions)#

Theorem: Continuity of Real Vector Functions

A real vector function \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) is continuous at \(\boldsymbol{p} \in \mathcal{D}\) if and only if \(\boldsymbol{p}\) is an isolated point of \(\mathcal{D}\) or \(f\) is equal to its own limit there:

Theorem: Continuity via Component Functions

A real vector function \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) is continuous at \(\boldsymbol{p} \in \mathcal{D}\) if and only if its component functions \(f_1, \dotsc, f_n\) are continuous at \(\boldsymbol{p}\).

Theorem: Continuity via Limits of Scalar Fields

A real vector function \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) is continuous at \(\boldsymbol{p} \in \mathcal{D}\) if and only if \(\boldsymbol{p}\) is an isolated point of \(\mathcal{D}\) or the limit of \(||f(\boldsymbol{x}) - f(\boldsymbol{p})||\) there is zero:

Theorem: Continuity of Linear Combination

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^m \to \mathbb{R}^n\) and \(g: \mathcal{D}_g \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be real vector functions.

If \(f\) and \(g\) are continuous at \(\boldsymbol{p} \in \mathcal{D}_f \cap \mathcal{D}_g\), then so is \(\lambda f + \mu g\) for all \(\lambda, \mu \in \mathbb{R}\).