Absolute Value Function is Even Function

Theorem
Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$:

Then $\size {\, \cdot \,}$ is an even function.

Proof
Recall the definition of the absolute value function:


 * $\size x = \begin{cases}

x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

Testing the $3$ cases in turn:

Hence the result by definition of even function.