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===Theorems regarding multivariate limits and continuity ===
* All properties of linearity and superposition from single-variable calculus carry over to multivariate calculus.
* '''Composition''': If <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> and <math>g: \mathbb{R}^m \to \mathbb{R}^p</math> are both multivariate continuous functions at the points <math>x_0 \in \mathbb{R}^n</math> and <math>f(x_0) \in \mathbb{R}^m</math> respectively, then <math>g \circ f: \mathbb{R}^n \to \mathbb{R}^p</math> is also a multivariate continuous function at the point <math>x_0</math>.
* '''Multiplication''': If <math>f: \mathbb{R}^n \to \mathbb{R}</math> and <math>g: \mathbb{R}^n \to \mathbb{R}</math> are both continuous functions at the point <math>x_0 \in \mathbb{R}^n</math>, then <math>fg: \mathbb{R}^n \to \mathbb{R}</math> is continuous at <math>x_0</math>, and <math>f/g : \mathbb{R}^n \to \mathbb{R}</math> is also continuous at <math>x_0</math> provided that <math>g(x_0) \neq 0</math>.
* If <math>f: \mathbb{R}^n \to \mathbb{R}</math> is a continuous function at point <math>x_0 \in \mathbb{R}^n</math>, then <math>|f|</math> is also continuous at the same point.
* If <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> is [[Lipschitz continuous]] (with the appropriate normed spaces as needed) in the neighbourhood of the point <math>x_0 \in \mathbb{R}^n</math>, then <math>f</math> is multivariate continuous at <math>x_0</math>.

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From the Lipschitz continuity condition for <math>f</math> we have

{{NumBlk|:|<math>|f(s(t))-f(s(t_0))| \leq K|s(t)-s(t_0)|</math>|{{EquationRef|8}}}}

where <math>K</math> is the Lipschitz constant. Note also that, as <math>s(t)</math> is continuous at <math>t_0</math>, for every <math>\delta > 0</math> there exists a <math>\epsilon > 0</math> such that <math>|s(t)-s(t_0)| < \delta</math> <math>\forall |t-t_0| < \epsilon</math>.

Hence, for every <math>\alpha > 0</math>, choose <math>\delta = \frac{\alpha}{K}</math>; there exists an <math>\epsilon > 0</math> such that for all <math>t</math> satisfying <math>|t-t_0| < \epsilon</math>, <math>|s(t)-s(t_0)| < \delta</math>, and <math>|f(s(t)) - f(s(t_0))| \leq K|s(t)-s(t_0)| < K\delta = \alpha</math>. Hence <math>\lim_{t \to t_0} f(s(t))</math> converges to <math>f(s(t_0))</math> regardless of the precise form of <math>s(t)</math>.

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