\( \operatorname{sgn} x=\left\{\begin{aligned}-1 & \text { if } x<0 \\ 0 & \text { if } x=0 \\ 1 & \text { if } x>0\end{aligned}\right. \)

The function you provided is known as the sign function, denoted as \( \operatorname{sgn}(x) \). It is defined piecewise as follows: \[ \operatorname{sgn} x = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \] ### Explanation: - If \( x \) is negative (i.e., \( x < 0 \)), the sign function returns \(-1\). - If \( x \) is zero (i.e., \( x = 0 \)), the sign function returns \(0\). - If \( x \) is positive (i.e., \( x > 0 \)), the sign function returns \(1\). ### Properties: 1. **Odd Function**: The sign function is an odd function, meaning that \( \operatorname{sgn}(-x) = -\operatorname{sgn}(x) \) for all \( x \). 2. **Discontinuity**: The function is discontinuous at \( x = 0 \). 3. **Range**: The range of the sign function is \(\{-1, 0, 1\}\). ### Applications: The sign function is often used in mathematics and engineering to determine the direction of a quantity, such as in optimization problems, control systems, and in defining piecewise functions.