ProofWiki:Sandbox

Theorem
Let $a \in \R_{>0}$.

Let $f : \Q \to \R$ be the real mapping defined as:
 * $f \left({ r }\right) = a^r$

where $a^r$ denotes $a$ to the power of $r$.

Then $f$ is monotone.

Further, $f$ is strictly monotone unless $a = 1$.

Case 1: $a > 1$
If $a > 1$, then $f$ is strictly increasing by  Exponential with Base Greater than One is Strictly Increasing.

Case 2 : $a = 1$
If $a = 1$, then $f$ is constant.

If $f$ is constant, then $f$ is both increasing and  decreasing from Mapping Constant iff Increasing and Decreasing.

Case 3: $0 < a < 1$
If $0 < a < 1$, then $f$ is strictly decreasing by  Exponential with Base Between Zero and One is Strictly Decreasing.

Hence the result.