I do not understand the following sentence:
One can see the Dirac-delta at $x$, $\delta(s-x)=\delta_x(s)$ as second derivative of $(s-x)_ {+} = \max(0,s-x)$ or $(x-s) _ +=\max(0,x-s)$ depending on if one wants the primitives zero to the left or to the right.
Can someone kindly elaborate on this ?
Notice that $(s-x)_+$ is a continuous function, $C^1$ except at $s=x$. Hence its derivative in the sense of distributions is the same a.e. as its pointwise derivative, that is $$ \partial_s (s-x)_+ = \mathbf 1_{(x,\infty)}(s) = \mathbf 1_{(0,\infty)}(s-x), $$ which is the function that is $0$ on $(-\infty,x)$ and $1$ on $(x,\infty)$. Taking once more its derivative in the sense of distributions gives since $\mathbf 1_{(0,\infty)}' = \delta_0$, $$ \partial_s^2 (s-x)_+ = \delta_{x}(s) = \delta_{0}(s-x). $$ Now the first point of the remark is that if you replace $s-x$ by $x-s$ it does not change the second derivative. Indeed, you will get $\partial_s (x-s)_+ = -\mathbf 1_{(-\infty,x)}(s) = -\mathbf 1_{(0,\infty)}(x-s)$, so here, it is different, there is an extra minus sign, but taking one more derivative cancels the minus sign by multiplying again by $-1$. Hence you again get $\partial_s^2 (x-s)_+ = \delta_{x}(s) = \delta_{0}(x-s)$.
Now one when doing the reverse process and taking primitives, the choice of the constant of integration will give different results. For example $$ \int_{-\infty}^t \delta_x(\mathrm d s) = \mathbf 1_{(x,\infty)}(t) $$ (this is a primitive that is $0$ on the left) while $$ \int_{\infty}^t \delta_x(\mathrm d s) = -\int_t^{\infty} \delta_x(\mathrm d s) = -\mathbf 1_{(-\infty,x)}(t) $$ (this is a primitive that is $0$ on the right). Integrating once more will give you $$ \int_{-\infty}^\tau\int_{-\infty}^t \delta_x(\mathrm d s)\,\mathrm d \tau = (\tau-x)_+ \\ \int_{\infty}^\tau\int_{\infty}^t \delta_x(\mathrm d s)\,\mathrm d t = \int_\tau^{\infty}\int_t^{\infty} \delta_x(\mathrm d s)\,\mathrm d t = (x-\tau)_+ $$ where the first function is $0$ on the left of $x$ while the second function is $0$ on the right of $x$.