\begin{definition}[Definition:O Notation/Big-O Notation/Real/Point]
Let $x_0 \in \R$.
Let $f$ and $g$ be real-valued or complex-valued functions defined on a punctured neighborhood of $x_0$.
The statement:
:$\map f x = \map \OO {\map g x}$ as $x \to x_0$
is equivalent to:
:$\exists c \in \R_{\ge 0}: \exists \delta \in \R_{>0}: \forall x \in \R : \paren {0 < \size {x - x_0} < \delta \implies \size {\map f x} \le c \cdot \size {\map g x} }$
That is:
:$\size {\map f x} \le c \cdot \size {\map g x}$
for all $x$ in a punctured neighborhood of $x_0$.
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
\end{definition}