ProofWiki:Sandbox

Theorem
The natural logarithm function is continuous.

Proof 1
Because the Natural Logarithm Function is Differentiable, the result follows from Differentiable Function is Continuous.

Proof 2
From Bounds of Natural Logarithm:
 * $\dfrac{1}{2} < \ln \left({ 2 }\right) < 1$

Fix $x \in \R$.

Consider $\dfrac{x}{ \ln \left({ 2 }\right) }$.

From Rationals are Everywhere Dense in Reals:
 * $\forall \epsilon \in \R_{>0} \exists r \in \Q : \left\vert{ r - \dfrac{x}{ \ln \left({ 2 }\right) } }\right\vert < \epsilon$

Thus:

Thus:
 * $\forall \epsilon \in \R_{>0} : \exists t \in \R : \left\vert{ \ln\left({ t } \right) - x }\right\vert < \epsilon$

Thus, the image of $\R_{>0}$ under $\ln$ is  everywhere dense in $\R$.

From Monotone Real Function with Everywhere Dense Image is Continuous, $\ln$ is continuous on $\R_{>0}$.

Hence the result.