\begin{definition}[Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1]
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 $X_1$.
'''$f$ is continuous at (the point) $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$)''' {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$
where $\R_{>0}$ denotes the set of all strictly positive real numbers.
\end{definition}