Definition:Continuous Real-Valued Vector Function

Definition
Let $\R^n$ be the cartesian $n$-space.

Let $f: \R^n \to \R$ be a real-valued function on $\R^n$.

Then $f$ is continuous on $\R^n$ :
 * $\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: \map d {x, a} < \delta \implies \size {\map f x - \map f a} < \epsilon$

where $\map d {x, a}$ is the distance function on $\R^n$:


 * $\ds d: \R^n \to \R: \map d {x, y} := \sqrt {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n}$ are general elements of $\R^n$.

Also see

 * Definition:Continuous Mapping (Metric Spaces)