Power Function on Base Greater than One is Strictly Increasing
Theorem
Natural Number
Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as:
- $\map f n = a^n$
where $a^n$ denotes $a$ to the power of $n$.
Then $f$ is strictly increasing.
Integer
Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Z \to \R$ be the real-valued function defined as:
- $\map f k = a^k$
where $a^k$ denotes $a$ to the power of $k$.
Then $f$ is strictly decreasing.
Rational Number
Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \Q \to \R$ be the real-valued function defined as:
- $\map f q = a^q$
where $a^q$ denotes $a$ to the power of $q$.
Then $f$ is strictly increasing.
Real Number
Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \R \to \R$ be the real function defined as:
- $\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is strictly increasing.