Definition:O Notation/Big-O Notation/Real/Point

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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:

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

is equivalent to:

$\exists c \in \R: c \ge 0: \exists \delta \in \R : \delta > 0 : \forall x \in \R : \left({0 < \left\lvert{x - x_0}\right\rvert < \delta \implies \left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert}\right)$


That is:

$\left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert$

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