Exponential Function is Well-Defined/Real/Proof 3

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then $\exp x$ is well-defined.

Proof
This proof assumes the continuous extension definition of $\exp$.

Let $e$ denote Euler's number.

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

From Euler's Number to Rational Power Permits Unique Continuous Extension, there exists a unique continuous extension of $f$ to $\R$.

Hence the result.