Definition:Continuous Mapping (Normed Vector Space)

This page is about Continuous Mapping in the context of Normed Vector Space. For other uses, see Continuous Mapping.

Definition

Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces.

Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$.

Let $a \in X_1$ be a point in $X_1$.

Continuous at a Point

$f$ is continuous at (the point) $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$) if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

Continuous on a Space

$f$ is continuous from $\struct{X_1, \norm {\,\cdot\,}_{X_1} }$ to $\struct{X_2, \norm {\,\cdot\,}_{X_2} }$ if and only if it is continuous at every point $x \in X_1$.