Exponential Function is Well-Defined/Real/Proof 4

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 Inverse of $\ln$ definition of $\exp$.

Let:
 * $k \in \N$
 * $J_k = \left[{ \dfrac{1}{k} \,.\,.\, k }\right]$.
 * $J_{k}^{o} = \left({ \dfrac{1}{k} \,.\,.\, k }\right)$.

and let $\ln_{ J_{k}^{o} }$ denote the restriction of $\ln$ to $J_{k}^{o}$.

Note that $\displaystyle \bigcup \left\{ \ln_{ J_{k}^{o} } : k \in \N \right\} = \ln$.

From Natural Logarithm Function is Continuous, $\ln$ is continuous on $J_k$.

Also:

From Derivative of Inverse Function, $\ln_{ J_{k}^{o} }$ is invertible.

From Union of Functions Theorem/Corollary, $\ln = \displaystyle \bigcup \left\{ \ln_{ J_{k}^{o} } : k \in \N \right\}$ is invertible.

Hence the result, from Inverse Mapping is Unique.