I just hate this function with a love.

Can somebody possibly explain to me how possibly it makes sense?

1)d(x) = 0 almost everywhere, then INT (-inf,inf)d(x)=0.

That is a theorem.

2)If L(f(x)) = L(g(x)) where L is Laplace transform then f(x)=g(x). Yet d(x) contradicts uniqueness.

Is the rule that mathemations just avoid this "function" and engineers just use it without and problems?