I have been reviewing some details on the Dirac Delta function and I've hit a little bit of a road block with trying to wrap my head around how the Translation/Sifting property of the function is justified. Now according to my text the overall definition is the generalized function with the "defining property" that for any continuous function f(x), the integral taken from negativ e infinity to infinity of the product of f(x) and the D-D function is the value of f(x) at x=0. (As also shown in eqn 1 of the attached file) Now a little ways down, the text introduces the "shifted" version of the integral (as shown in eqn 2), and says that through a change of variable to u=x-a, it casts the integral into the form of eqn 1. However, I see no justification algebraically for this step. Intuitively I see it as just shifting the "focus" of the Dirac-Delta function to pick out f's value at x=a rather than x=0, but am looking to see a formal justification of this. In short basically what I'm wondering is how do I go from having the defining integral as in equation 1 to the integral in equation 2. I would ask for just how the "substitute variable" approach is justified, but if you can provide an explanation that's more rigorous, that would be much appreciated. Also, I have to apologize. As I am not familiar with Latex, I have resorted to trying to use a paint utility to write out the equations. I have tried my best to represent them, but am no artist.