The Signum Function — Cheat sheet

CHEAT SHEET · FREE

Key method

\operatorname{sgn}(x)=\begin{cases}-1,&x<0,\\0,&x=0,\\1,&x>0.\end{cases}

For $x\ne0$ , it equals $x/|x|$ . At zero, that quotient is undefined, so the separate definition is essential.

Example

The left-hand limit at zero is $-1$ and the right-hand limit is $1$ . Consequently, no two-sided limit exists there, even though the function value $0$ is defined. Away from zero, the function is locally constant and its derivative is zero.

Avoid this mistake

Defining a value at a jump cannot make the two one-sided limits equal. The derivative of $|x|$ is signum only for $x\ne0$ ; $|x|$ is not differentiable at zero.