Exponential Function is Continuous/Real Numbers/Proof 4

Theorem

The real exponential function is continuous.

That is:

$\forall x_0 \in \R: \ds \lim_{x \mathop \to x_0} \exp x = \exp x_0$

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.

$\blacksquare$