ProofWiki:Sandbox

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.

Fix $x_0 \in \R$.

Consider $I := \left [{x_0 - 1 \,.\,.\, x_0 + 1} \right]$.

From Closed Bounded Subset of Real Numbers is Compact, $I$ is compact.

Let $f | \Q \to \R$ be the mapping defined by $f \left({ x }\right) = e^x$.

From Exponential is Continuous on the Rationals, $f$ is continuous on $\Q$.

From Continuous Mapping is Uniformly Continuous on Compact Space, $f$ is uniformly continuous on $I \cap \Q$.

From Rationals are Everywhere Dense in Reals, $I \cap \Q$ is everywhere dense in $I$.

From Continuous Extension from Dense Subset, there exists a unique continuous extension of $f$ to $I$.

In particular, $f$ is continuous at $x_0$.