Definition:Exponential Function/Real/Extension of Rational Exponential

Definition
Let $e$ denote Euler's number.

Let $f: \Q \to \R$ denote the real-valued function defined as:
 * $f \left({ x }\right) = e^x$

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

 * Equivalence of Definitions of Real Exponential Function