Definition:Power (Algebra)/Real Number/Definition 2
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Definition
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $r \in \R$ be a real number.
Let $f : \Q \to \R$ be the real-valued function defined as:
- $f \left({ q }\right) = x^q$
where $a^q$ denotes $a$ to the power of $q$.
Then we define $x^r$ as the unique continuous extension of $f$ to $\R$.
Also see
- Power Function to Rational Power permits Unique Continuous Extension, where the existence and uniqueness of such an extension is proven.
Sources
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction