We have a differential equation:
$$a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = \delta(x-z) $$
And this is satisfied by the Green's function $G(x,z)$. I've understood this far. However, I don't understand the following paragraph (Riley Hobson and Bence, Mathematical methods for Physics and Engineering):
My questions are as follows:
Why is $\frac{d^n G(x,z)}{dx^n}$ infinity at $x = z$?
Why are all the derivatives of order less than $n-1$ continuous?
Thank you.