Graph of the Sign Function $y = \operatorname{sgn} x$

$y = \operatorname{sgn} x$ is the sign (signum) function, which extracts only the sign of a real number $x$ and returns one of three values.

It returns $1$ when $x$ is positive, $-1$ when negative, and $0$ only when $x$ is exactly $0$.

• $\operatorname{sgn}(3.5) = 1$
• $\operatorname{sgn}(-2) = -1$
• $\operatorname{sgn}(0) = 0$

Domain and range. The domain is all real numbers, but the range consists of just the three values $\{-1,\ 0,\ 1\}$. The graph is two horizontal rays — height $-1$ for $x < 0$ and height $1$ for $x > 0$ — with a single isolated point of height $0$ at the origin.

Symmetry. Since flipping the sign of $x$ flips the value, $\operatorname{sgn}(-x) = -\operatorname{sgn} x$, the function is odd (symmetric about the origin). The value $\operatorname{sgn}(0) = 0$ is consistent with this.

Discontinuities and jumps. The only discontinuity is at the origin $x = 0$. The left-hand limit is $-1$ and the right-hand limit is $+1$, so they disagree and the jump has size $2$; the value itself is the midpoint $0$. Everywhere else the function is constant and therefore continuous.

Relation to other functions. The sign function is closely tied to the absolute value:

The right-hand formula only works for $x \neq 0$. Differentiating $|x|$ gives $\operatorname{sgn} x$ (for $x \neq 0$), and the function relates to the Heaviside step $H(x)$ by $\operatorname{sgn} x = 2H(x) - 1$.

Applications. The sign function is handy for expressing 'direction' or a positive/negative test in a formula. It appears in absolute-value calculations, in on-off (bang-bang) control, and in sign-based classification in machine learning — anywhere the direction of a value matters more than its magnitude.