Delta sequence "extrapolation"

[Elm8429]

TL;DR Summary

Studying delta sequences and am wondering if $\phi_n(x)$ is a delta sequence, does it extends that property to all of it's arguments ?

Studying delta sequences and am wondering if $\phi_n(x)$ is a delta sequence (it resembles and validates the sifting property), will $\phi_n(x - a)$ or $\phi_n(whatever)$ respect the sifting property ? I think I get confused in the semantics, because reading my textbook, I get the following where I test if $\phi_n(x)$ is a delta sequence if it has the expected sifting property of the delta function, but then we use $\phi_n(x)$ to test all the other sifting properties of the delta function for different arguments ! : we confirmed that $\phi_n(x)$ is a delta sequence, so $\phi_n(whatever)$ can express the delta function of $\delta(whatever)$ the get "whatever" shifting property.

I hope my confusion was clear and that someone will remedy my theoretical woes !

 

[anuttarasammyak]

Though I do not know the context, say distribution function for continuous variable x
$$\phi_n(x)=\delta(x-a_n)$$,
$$\int f(x) \phi_n(x)dx =f(a_n)$$
for any f(x).