# Odd Power Function is Strictly Increasing

## Theorem

### Real Numbers

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

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

$\map {f_n} x = x^n$

Then $f_n$ is strictly increasing.

### General Result

Let $\struct {R, +, \circ, \le}$ be a totally ordered ring.

Let $n$ be an odd positive integer.

Let $f: R \to R$ be the mapping defined by:

$\map f x = \map {\circ^n} x$

Then $f$ is strictly increasing on $R$.