Definition:Big-O Notation/General Definition/Point

Definition

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 $x_0$.