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[parent] formal definition of Landau notation (Definition)

Let us consider a domain $ D$ and an accumulation point $ x_0\in \overline D$. Important examples are $ D=\mathbb{R}$ and $ x_0\in D$ or $ D=\mathbb{N}$ and $ x_0=+\infty$. Let $ f\colon D\to \mathbb{R}$ be any function. We are going to define the spaces $ o(f)$ and $ O(f)$ which are families of real functions defined on $ D$ and which depend on the point $ x_0\in \overline D$.

Suppose first that there exists a neighbourhood $ U$ of $ x_0$ such that $ f$ restricted to $ U\cap D$ is always different from zero. We say that $ g\in o(f)$ as $ x\to x_0$ if

$\displaystyle \lim_{x\to x_0} \frac{g(x)}{f(x)}=0. $
We say that $ g \in O(f)$ as $ x\to x_0$ if there exists a neighbourhood $ U$ of $ x_0$ such that
$\displaystyle \frac{g(x)}{f(x)}$   is bounded if restricted to $ D\cap U$$\displaystyle . $
In the case when $ f\equiv 0$ in a neighbourhood of $ x_0$, we define $ o(f)=O(f)$ as the set of all functions $ g$ which are null in a neighbourhood of 0.



"formal definition of Landau notation" is owned by paolini.
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See Also: properties of $O$ and $o$

Other names:  Landau notation, small o, big o, order of infinity, order of zero

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Cross-references: null, restricted, neighbourhood, point, real functions, function, accumulation point, domain
There are 8 references to this entry.

This is version 2 of formal definition of Landau notation, born on 2005-05-12, modified 2005-05-16.
Object id is 7049, canonical name is FormalDefinitionOfLandauNotation.
Accessed 4867 times total.

Classification:
AMS MSC: 26A12 (Real functions :: Functions of one variable :: Rate of growth of functions, orders of infinity, slowly varying functions)
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