Exponential Function is Continuous/Real Numbers/Proof 2

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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 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.

$\blacksquare$