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.
$\blacksquare$