Exponential Function is Continuous/Real Numbers/Proof 4

Theorem
The real exponential function is continuous on $\R$.

That is:


 * $\forall x_0 \in \R: \displaystyle \lim_{x \to x_0} \ \exp x = \exp x_0$

Proof
This proof depends on the continuous extension definition of the exponential function.

Let $\exp$ be the unique continuous extension of $e^x$ from $\Q$ to $\R$.

By definition, $\exp$ is continuous.

Hence the result.