Definition:Sufficiently Small

Definition
Let $P$ be a property of real numbers.

We say that $P \left({x}\right)$ holds for all sufficiently small $x$ :


 * $\exists \epsilon \in \R: \forall x \in \R: |x| \le \epsilon: P \left({x}\right)$

That is:
 * There exists a real number $\epsilon$ such that the for every (real) number not more than $\epsilon$ in in absolute value, the property $P$ holds.

It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $\epsilon$ is actually known, just that such a value exists.

Also known as
To avoid possible ambiguity, this concept is sometimes called:


 * "sufficently small in absolute value"

or


 * "sufficiently small in magnitude

The ambiguity to be avoided is that in natural language, one might refer to $-x$ as "small" if $x > 0$ is "large".

Also see

 * Definition:Sufficiently Large
 * Definition:Arbitrarily Small