Definition:Exponential Function/Real/Extension of Rational Exponential

Definition
Let $e$ denote Euler's number.

Let $f : \Q \to \R$ denote the mapping defined by $f \left({ x }\right) = e^x$.

That is, let $f \left({ x }\right)$ 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 $\R$.

$\exp \left({ x }\right)$ is called the exponential of $x$.

Also see

 * Exp x equals e^x for Rational Numbers