Power Function on Base Greater than One is Strictly Increasing

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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.