Rational Power is of Exponential Order Epsilon

Theorem
Let $r = \dfrac p q$ be a rational number, with $p, q \in \Z: q \ne 0, r > 0$.

Then:


 * $t \mapsto t^r$

is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.

Proof
For $t > 0$, $t^r$ is continuous.

At $t = 0$, the discontinuity is removable:

Write $t^r = t^{p/q}$, and set $t > 1$.

Recall from Polynomial is of Exponential Order Epsilon, $t^p < K'e^{a't}$ for any $a' > 0$, arbitrarily small in magnitude.

Therefore the inequality $\ t^{p/q} < Ke^{at}$ has solutions of the same nature.