Definition (Dirac delta function)

Heuristically, we refer to the Dirac delta function as the function that we integrate over to get the distribution above which can be loosely thought of as $$\delta(t)=\begin{cases} \infty & t=0\\0 & t\neq 0 \end{cases}$$giving us $$\int\limits _{-\infty}^{\infty}\delta(x) \, dx =1$$