The Signum Function

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Idea

The signum function records only the sign of an input: negative, zero, or positive. It discards magnitude. This makes it useful for separating direction from size.

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.

Worked 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.

Common 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.

Check your understanding

What is $\operatorname{sgn}(-7)+\operatorname{sgn}(0)+\operatorname{sgn}(2)$ ?

Show answer

$-1+0+1=0$ .

Explore

Domain and range

Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.

Function

sgn(x) = −1 for x < 0, 0 at x = 0, 1 for x > 0 · Range: {−1, 0, 1}

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is sgn(−7)?

−1

−7

Hint 1 · Find a starting point

The signum function reports sign, not magnitude.

Hint 2 · Take the next step

Every negative input gets the same output.

Show the reasoning

Answer: −1

sgn(x)=−1 for x<0, 0 at zero and 1 for x>0.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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