Absolute Value is Functional

Theorem

Let $f: \R \to \R$ be the absolute value function:

$\forall x \in \R: f \left({x}\right) = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases}$

Then $f$ is a functional relation.

Proof

Let $f \left({x_1}\right) = y_1, f \left({x_2}\right) = y_2$ where $y_1 \ne y_2$.

The result follows.

$\blacksquare$

Sources

• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Exercise $1$