Exponential Function is Continuous

Theorem
The exponential function is continuous. That is,


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

Proof 1
This proof depends on the limit definition of the exponential function.

Because the exponential function is defined as a limit, the result follows from the definition of continuity.

Proof 2
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.

Proof 3
This proof depends on the differential equation definition of the exponential function.

The result follows from Differentiable Function is Continuous.