Question on Dirac Delta function in Griffiths

My question is in Griffiths Introduction to Electrodynamics 3rd edition p48. It said

Two expressions involving delta function ( say ##D_1(x)\; and \;D_2(x)##) are considered equal if:
[tex]\int_{-\infty}^{\infty}f(x)D_1(x)dx=\int_{-\infty}^{\infty}f(x)D_2(x)dx\;[/tex]6
for all( ordinary) functions f(x).

In note 6 at the bottom of the page:
This is not as arbitrary as it may sound. The crucial point is that the integral must be equal for any f(x). Suppose ##D_1(x)\; and \; D_2(x)## actually differed, say, in neighborhood of the point x=17. Then we could pick a function f(x) that was sharply peaked about x=17, and the integrals would not be equal.




I don't understand what the book is talking about. You have two different Dirac Delta functions and can be equal for all f(x).

Say ##D_1(x)=\delta (x-2) \; and \; D_2(x)=\delta(x-4)##. Let f(x)=x:
[tex]\int_{-\infty}^{\infty}x\delta (x-2)dx=f(2) \int_{-\infty}^{\infty}\delta (x-2)dx=2[/tex]
[tex]\int_{-\infty}^{\infty}x\delta (x-4)dx=f(4) \int_{-\infty}^{\infty}\delta (x-2)dx=4[/tex]

How is these two equal?

Thanks