Odd Power Function is Strictly Increasing
Jump to navigation
Jump to search
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 $R$.