Exponential Function is Continuous/Real Numbers/Proof 5

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 series definition of $\exp$.

That is, let:

$\ds \exp x = \sum_{k \mathop = 0}^ \infty \frac {x^k} {k!}$

From Series of Power over Factorial Converges, the radius of convergence of $\exp$ is $\infty$.

Thus, from Power Series Converges to Continuous Function, $\exp$ is continuous on $\R$.

$\blacksquare$