Odd Power Function is Strictly Increasing/Real Numbers

Theorem
Let $n \in \Z_{> 0}$ be an odd positive integer.

Let $f_n: \R \to \R$ be the real function defined as:


 * $f_n \left({x}\right) = x^n$

Then $f_n$ is strictly increasing.

Proof
From the Power Rule for Derivatives, $D_x \left({x^n}\right) = n x^{n-1}$.

As $n$ is odd, $n-1$ is even.

Thus by Even Power is Non-Negative, $D_x \left({x^n}\right) \ge 0$ for each $x$.

From Derivative of Monotone Function, it follows that $f_n$ is increasing over the whole of $\R$.

The only place where $D_x \left({x^n}\right) = 0$ is at $x = 0$.

Everywhere else, $f_n$ is strictly increasing.

By Sign of Odd Power,
 * $f_n \left({x}\right) < 0 = f_n \left({0}\right)$ when $x < 0$ and
 * $f_n \left({0}\right) = 0 < f_n \left({x}\right)$ when $0 < x$.

Thus $f_n$ is strictly increasing on $\R$.