Power Function is Monotone/Rational Number

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

Let $f : \Q \to \R$ be the real-valued function 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: $a > 1$
Let $a > 1$.

Then by Power Function on Base Greater than One is Strictly Increasing:


 * $f$ is strictly increasing.

By Strictly Increasing Mapping is Increasing:


 * $f$ is increasing.

Case 2 : $a = 1$
Let $a = 1$.

Then:
 * $\forall r \in \Q: a^r = 1$

and so $f$ is constant.

From Mapping Constant iff Increasing and Decreasing:
 * $f$ is both increasing and decreasing.

Case 3: $0 < a < 1$
Let $0 < a < 1$.

Then by Power Function on Base between Zero and One is Strictly Decreasing:


 * $f$ is strictly decreasing.

By Strictly Decreasing Mapping is Decreasing:


 * $f$ is decreasing.

Conclusion
It has been shown that in all cases $f$ is either increasing or decreasing or both.

Thus by definition of monotone function, $f$ is monotone.

In either of the cases where $a \ne 1$, It has been shown that in all cases $f$ is either strictly increasing or strictly decreasing.

Thus by definition of strictly monotone function, $f$ is strictly monotone.

Hence the result.