I've been working on this problem set for a little bit now and I've finally made it to the last question. I'm now left with this monster and I don't quite understand where to proceed. All the previous problems were much simpler functionals and their derivatives were easy to calculate.
With this functional I don't really understand how I am supposed to treat $\Delta(x-y)$ when using the definition of the functional derivative to caclulate $\large\frac{\delta\mathcal{Z_0}[J]}{\delta J(\mathcal{z_1})}$. Any sort of clarification about that specific delta function or any trick that I could use to calculate this would be much appreciated.
Instead of you, I would study in terms of $H$ the difference $Z_0 (J+H) - Z_0 (J)$, because the term in $H$ that would appear would be exactly the differential at $J$ evaluated at $H$, and you'll be done.