Exponential Function is Continuous/Real Numbers/Proof 5
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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 $\R$.
$\blacksquare$