Symmetry of delta functions

Question

Proving that the delta function is symmetric

The above link has answers for the symmetry of the delta function as:

$V(x)=δ(x)$

If suppose I have a

$V(x)=λ(δ(x-ap)+δ(x-aq))$

And

$V(x)=λ(δ(x-a)+δ(x+a))$

I would guess the former is not even and the second is even. Can someone confirm it? Thanks in advance.

Answer 1

All should be even as delta function is even as

$δ(x)=δ(-x)$

Which implies,

$δ(x-a)=δ(-(x-a))$

And further,

$δ(x-a)+δ(x-b)=δ(-(x-a))+δ(-(x-b))$

The same can be shown using definition of delta functions as done in one of the answers for the above mentioned post in the question.