Definition:Big-O Notation/General Definition/Point

Definition
Let $(X,\tau)$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$

Let $x_0\in X$.

Let $f,g:X\setminus\{x_0\}\to V$ be functions.

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

is equivalent to:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists U\in\tau: x_0\in U : \forall x\in U\setminus\{x_0\} : \Vert f(x)\Vert \leq c\cdot\Vert g(x)\Vert$

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

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