nLab
sign function

Contents

Idea

The sign function on the real numbers.

Definition

sgn x {1 | x>0 0 | x=0 1 | x<0 \array{ \mathbb{R} &\overset{sgn}{\longrightarrow}& \mathbb{R} \\ x &\mapsto& \left\{ \array{ 1 &\vert& x \gt 0 \\ 0 &\vert& x = 0 \\ -1 &\vert& x \lt 0 }\right. }

Some definitions will modify the value at 00, usually to make it 11, 1-1, or undefined. In many applications, the sign function is essentially treated as a measurable function on the real line with Lebesgue measure, and then these are all essentially the same since they are all almost equal. But some applications require the function to be left-? or right-continuous?, and then the value of 00 must be chosen appropriately, while the handy formula x/|x|x/{|x|} naturally makes the value at 00 undefined.

Related concepts

References

See also

Last revised on August 22, 2020 at 21:02:02. See the history of this page for a list of all contributions to it.