Absolute Function for Real Numbers

Definition

$$|x|=\sqrt{x^2}$$ $$|x|=\{x,\text{if }x\ge 0,-x\text{if }x<0$$

To Piecewise

Properties

1. $|x|=\sqrt{x^2}$
2. $|x|\ge 0$
3. $|-x|=|x|$
4. $-|x|\le x\le |x|$
5. $|xy|=|x||y|$
6. Triangle Inequality
7. Reverse Triangle Inequality

Limits

$|x|$ For a function like this, there is NO limit. Because, it is a piecewise function And so ${\lim }_{x\to a^{-}}$ will not be the same as ${\lim }_{x\to a^{+}}$

Not Differentiable at 0

Cusp discontinuity. $f(x)=|x|$ $f^{\prime }(0^{-})=-1$ $f^{\prime }(0^{+})=1$ -1 $\ne$ 1