Definition:Big-O Notation/Real/Point

Definition

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 $x_0$.