Exponential Function is Continuous/Real Numbers/Proof 2

Proof
This proof depends on the definition of the exponential function as the function inverse of the natural logarithm.

From Logarithm is Strictly Increasing, $\ln$ is strictly monotone on $\R_{>0}$.

From Real Natural Logarithm Function is Continuous, $\ln$ is continuous on $\R_{>0}$

Thus, from the Continuous Inverse Theorem, $\exp := \ln^{-1}$ is continuous.