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:
 * $f(x) = \mathcal O \left({g(x)}\right)$ as $x \to x_0$

is equivalent to:
 * $\displaystyle \exists c \in \R: c\ge 0 : \exists \delta \in \R : \delta > 0 : \forall x \in \R : (0 < |x - x_0| < \delta \implies |f(x)| \leq c \cdot |g(x)|)$

That is:
 * $|f(x)| \leq c \cdot |g(x)|$

for all $x$ in a punctured neighborhood of $x_0$.