# Definition:Continuous Real-Valued Vector Function

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## 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 $\R^n$** iff:

- $\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: d \left({x, a}\right) < \delta \implies \left|{f \left({x}\right) - f \left({a}\right)}\right| < \epsilon$

where $d \left({x, a}\right)$ is the distance function on $\R^n$:

- $\displaystyle d: \R^n \to \R: d \left({x, y}\right) := \sqrt {\left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)}\right)}$

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

## As a Metric Space

Note that the definition for continuity as given here is the same as that for a metric space, where the Euclidean metric is taken on the real cartesian space.