Euler's Number to Rational Power permits Unique Continuous Extension

Theorem
Let $e$ be Euler's number.

Let $f: \Q \to \R$ be the real-valued function defined as:
 * $f \left({q}\right) = e^q$

where $e^q$ denotes $e$ to the power of $q$.

Then there exists a unique continuous extension of $f$ to $\R$.

Proof
Since $e > 0$, we may apply Power Function to Rational Power permits Unique Continuous Extension.

Hence the result.