Exponential Function is Continuous/Real Numbers/Proof 1
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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 limit definition of the exponential function.
Let:
- $\ds \exp x = \lim_{n \mathop \to \infty} \paren {1 + \dfrac x n}^n$
Fix $x_0 \in \R$.
Consider $I := \closedint {x_0 - 1} {x_0 + 1}$.
From Closed Bounded Subset of Real Numbers is Compact, $I$ is compact.
From Exponential Sequence is Uniformly Convergent on Compact Sets:
- $\paren {1 + \dfrac x n}^n$ is uniformly convergent on $I$.
By the Uniform Limit Theorem:
- $\ds \lim_{n \mathop \to \infty} \paren {1 + \dfrac x n}^n = \exp x$
is continuous on $I$.
In particular, $\exp x$ is continuous at $x_0$.
$\blacksquare$