Old unsolved question on greens function So I was looking up Kf Riley’s 3rd edition, and bump into a problem about greens function. I went online and googled and notice other people had the same problem and no one really could answer:
Continuity of Green's function and its derivatives https://math.stackexchange.com/questions/2403134/continuity-of-greens-function-and-its-derivatives
Continuity in derivatives of the Green's function in RHB
https://math.stackexchange.com/questions/3173174/continuity-in-derivatives-of-the-greens-function-in-rhb?rq=1
I suspect this is likely as there is some “hand waving” involve. Otherwise, why would the nth derivative tend to infinity, (n-1) derivative be discontinuous and (n-2) onwards is continuous? 
Is there some sort of intuition behind it? If not, why didn’t the authors devote some time to explaining it?
Any reference or direction appreciated
 A: This is not really handwaving but has been made precise. The point is thate you are really looking at distributions, not functions. After all, the Greens function is a solution of
$$ D\,G(x)=\delta(x)\;,$$
where $D$ is the differential operator for the problem. In physics applications $D$ is often the Laplace or Klein-Gordon operator. Even though physicists sometimes refer to $\delta$ as $\delta$-function, $\delta$ is indeed a distribution, and so is $G$. To have a real justification you need to look at some intro to the theory of functions. But of course you can get a feeling for what's going on by recalling that the function $f(x)=|x|$ is smooth,  $f'(x)=2\theta(x)-1$ and therefore $f''(x)=2\delta(x)$. Now you are looking for a $G$ that satisfies $\partial_x^nG(x)\sim\delta(x)$, where $n$ is the order of $D$. The above example of $f$, $f'$ and $f''$ then suggests that if $\partial_x^nG(x)$ has infinites, i.e. goes like $\delta$, then $\partial_x^{n-1}G(x)$ should go like $\theta$, i.e. jump, and $\partial_x^{n-2}G(x)$ like $|x|$, i.e. be continuous. These statements can be made precise. Of course you need to make certain assumptions on the test functions, they are usually taken to be either in $C^\infty$ with compact support or in the Schwartz space. If you choose to use freaky test functions, the above statements may not hold. 
