Integer Square Function is not Injective

Example of Mapping which is Not an Injection
Let $f: \Z \to \Z$ be the real square function:
 * $\forall x \in \Z: \map f x = x^2$

Then $f$ is not an injection.

Proof
For $f$ to be an injection, it would be necessary that:
 * $\forall x_1, x_2 \in \R: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$

By definition of the squaring operation, we have:
 * $\map f x = \map f {-x}$

But unless $x = 0$ it is not the case that $x = -x$.

Hence $f$ is not an injection.