\begin{definition}[Definition:Power (Algebra)/Real Number/Definition 2]
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 $\R$.
\end{definition}