Delta functions are confusing

deltafunc(x_0 - x_1)

 
 
source : http://www.physicspages.com/2014/11/08/functionals-and-functional-derivatives/

What does this mean? First, it's like the derivative rule for polynomials except the second derivative creates this delta function between two variables. What does that even mean? When two variables have the same value, the integral of the delta function equals 1?

Q2: Second, the result of the first differentiation is not another functional, it is a function, so how can we differentiate it as a functional?

A2: A function can be thought of as a trivial functional:

So the two variabled delta function is saying, pick out the function 6 f(x0) when integrated and x1=x0.... ok what monstrosity of mathematics is that pseudo-function?

delta function is ... what is that term? f(a) = f(-a), you can switch the x and the x0. d(x - x0) = d(x0 - x)