Review of the Dirac delta function

The delta function is defined as Where, the function f(x) is an arbitrary continuous function. From the definition, the following simple properties follow immediately: The delta function is obtained as a limit of functions. Below, we list three well-known examples. First, the step: The Dirac delta function is obtained by taking the ϵ→0 at the end of the integration. The Gaussian representation: The Lorentzian representation: Another representation of the delta function is the following integral: To prove this we write: which is the Lorentzian representation of the delta function with ϵ=λ/(2π)⁠. Notice that the integral and the λ→0+ limit do not commute. Namely, the integral with λ=0 is undefined. Show that the Gaussian and the Lorentzian in the limit of ϵ→0 satisfy all the properties of the Dirac delta function.