Power Function is Monotone/Rational Number

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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 $a = 1$.


Proof

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 is 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.

$\blacksquare$