Absolute Function

analysis

Definition

Absolute Function

The absolute function (or absolute value function) $f:\mathbb{R}\to \mathbb{R}$ is defined by:

$$f(x)=|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$$

It returns the non-negative distance of a real number from zero on the number line.

Properties

• Non-negativity: $|x|\ge 0$ for all $x\in \mathbb{R}$.
• Identity of Indiscernibles: $|x|=0⟺x=0$.
• Multiplicativity: $|xy|=|x||y|$.
• Triangle Inequality: $|x+y|\le |x|+|y|$.
• Continuity: The absolute function is continuous everywhere but not differentiable at $x=0$.