Signum Function¶

Definition¶

The signum function $\operatorname{sgn} \colon \mathbb{R} \to \mathbb{R}$ is defined by

\operatorname{sgn}(x)=\begin{cases} \displaystyle \, \frac{|x|}{x},& x\neq 0,\\[2pt] \displaystyle \, 0,& x= 0 \end{cases} =\begin{cases} \displaystyle \, 1,& x>0,\\[2pt] \displaystyle \, 0,& x=0,\\[2pt] \displaystyle \, -1,& x<0. \end{cases}

The signum function: a horizontal line at 1 for positive x, a horizontal line at -1 for negative x, and the value 0 at the origin.

Interpretation¶

The signum function keeps only the sign of a real number and discards its size: every positive number goes to $1$ , every negative number to $-1$ , and $0$ to $0$ . The formula $|x|/x$ shows the link with the absolute value: dividing a nonzero number by its magnitude leaves only its direction on the real line.

Read the other way, every real number factors as $x=\operatorname{sgn}(x)\,|x|$ $sgn (x) ∣ x ∣$ , sign times magnitude, and this also holds at $x=0$ $0$ . The three possible values match the trichotomy of real numbers: each $x$ is positive, zero or negative.

The signum function is the standard example of a jump. Its one-sided limits at $0$ are $1$ from the right and $-1$ from the left, so the limit at $0$ does not exist and no choice of $\operatorname{sgn}(0)$ makes the function continuous there. It also shows why the intermediate value theorem needs continuity: on $[-1,1]$ it never takes the value $\tfrac{1}{2}$ .

A common slip is to use $|x|/x$ at $x=0$ $0$ . That quotient is undefined at $0$ , which is why the definition sets $\operatorname{sgn}(0)=0$ separately.

Key Results¶

The signum function takes values in $\{-1,\,0,\,1\}\text{.}$
It reflects the trichotomy of real numbers.
The function is discontinuous at zero.

Examples¶

$\operatorname{sgn}(3)=1\text{.}$
$\operatorname{sgn}(-2)=-1\text{.}$
$\operatorname{sgn}(0)=0\text{.}$

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 5

Aliases: sign function, sgn Symbols: real numbers

Last updated: 22 October 2026 19:04