\begin{definition}[Definition:Dirac Delta Function/Definition 1]
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
Consider the real function $F_\epsilon: \R \to \R$ defined as:
:$\map {F_\epsilon} x := \begin{cases}
0 & : x < 0 \\
\dfrac 1 \epsilon & : 0 \le x \le \epsilon \\
0 & : x > \epsilon \end{cases}$
The '''Dirac delta function''' is defined as:
:$\map \delta x := \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
\end{definition}