Exponential Function is Continuous
Contents |
Theorem
The exponential function is continuous.
That is:
- $\forall c \in \C: \displaystyle \lim_{x \to c} \ \exp x = \exp c$
The scope of this needs to be expanded for the complex plane. This will not be possible till the basics of complex analysis have been properly addressed.
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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.
$\blacksquare$
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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.
$\blacksquare$
Proof 3
This proof depends on the differential equation definition of the exponential function.
The result follows from Differentiable Function is Continuous.
$\blacksquare$
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