\begin{definition}[Definition:Exponential Function/Real/Extension of Rational Exponential]
Let $e$ denote Euler's number.
Let $f: \Q \to \R$ denote the real-valued function defined as:
:$\map f x = e^x$
That is, let $\map f x$ denote $e$ to the power of $x$, for rational $x$.
Then $\exp : \R \to \R$ is defined to be the unique continuous extension of $f$ to $\R$.
$\map \exp x$ is called the '''exponential of $x$'''.
\end{definition}