\begin{definition}[Definition:O Notation/Big-O Notation/General Definition/Point]
Let $\struct {X, \tau}$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.
Let $x_0 \in X$.
Let $f, g: X \setminus \set {x_0} \to V$ be functions.
The statement
:$\map f x = \map \OO {\map g x}$ as $x \to x_0$
is equivalent to:
:$\exists c \in {\R}_{\ge 0}: \exists U \in \tau: x_0 \in U: \forall x \in U \setminus \set {x_0}: \norm {\map f x} \le c \norm {\map g x}$
That is:
:$\norm {\map f x} \le c \norm {\map g x}$
for all $x$ in a punctured neighborhood of $x_0$.
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
\end{definition}