Exponential Function is Continuous/Real Numbers/Proof 2

Theorem
The real exponential function is continuous.

That is:


 * $\forall x_0 \in \R: \displaystyle \lim_{x \to x_0} \ \exp x = \exp x_0$

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

Then the result follows from the continuity of inverse functions.