Euler's Number to Rational Power permits Unique Continuous Extension
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Theorem
Let $e$ be Euler's number.
Let $f: \Q \to \R$ be the real-valued function defined as:
- $f \left({q}\right) = e^q$
where $e^q$ denotes $e$ to the power of $q$.
Then there exists a unique continuous extension of $f$ to $\R$.
Proof
Since $e > 0$, we may apply Power Function to Rational Power permits Unique Continuous Extension.
Hence the result.
$\blacksquare$