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

$y = \operatorname{sgn} x$ is called the sign function. It extracts nothing but the sign of a real number $x$ and returns one of three values¹.

Domain and range

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

Symmetry

Since $\operatorname{sgn}(-x) = -\operatorname{sgn} x$, the function is odd, with point symmetry about the origin. The value $\operatorname{sgn}(0) = 0$ is consistent with that symmetry; indeed taking $0$ at the origin is forced by oddness.

Discontinuity and jump

The origin is the only point of discontinuity. The limit from the left is $-1$ and from the right is $+1$, so the two disagree and the jump has size $2$. The value itself is $0$, halfway between. Everywhere else the function is constant, hence continuous, with derivative $0$.

Relation to the absolute value

The sign function is closely tied to the absolute value.

The second holds only for $x \neq 0$. Differentiating $|x|$ gives $\operatorname{sgn} x$ on $x \neq 0$: differentiating a function with a corner produces a function with a jump.

Relation to the step function

It is related to the Heaviside step function $H(x)$ as follows².

The step function takes the two values $0$ and $1$; doubling it vertically and lowering it by $1$ gives the sign function. The range then becomes symmetric between $-1$ and $1$, which makes it convenient for representing a direction.

Applications

• Computing an absolute value, and folding a case analysis into a single formula
• On-off control in control engineering
• Classification by sign in machine learning

It comes into its own wherever the direction of a quantity matters more than its size.

1. Sign function, Wikipedia
2. Heaviside step function, Wikipedia