Mapping/Examples/x^3-x on Real Numbers

Example of Mapping
Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:
 * $\forall x \in \R: \map f x = x^3 - x$

Then $f$ is a surjection but not an injection.

Proof
Let $y \in \R$.

As $x \to \infty$, we have that $y \to \infty$.

Similarly, as $x \to -\infty$, we have that $y \to -\infty$.

From Real Polynomial Function is Continuous, $f$ is continuous on $\R$.

It follows from the Intermediate Value Theorem that:
 * $\forall y \in \R: \exists x \in \R: y = \map f x$

Thus, by definition, $f$ is a surjection.

We have that:

demonstrating that $f$ is not an injection.