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.

$\blacksquare$