 Neumann Green function for the modified Helmholtz equation on an interval

For real $\kappa \ne 0$, this Green function inverts $d^2/d{x}^2-{\kappa }^2$ on $[0,1]$ with homogeneous Neumann boundary conditions. It is continuous and its first derivative has jump one at $x=\xi$. At $\kappa =0$, a Green function with the same unit Dirac delta source and zero boundary derivatives does not exist: integrating its defining equation would give zero equal to one. The equation $y^{\prime \prime }=f$ with these boundary conditions instead requires ${\int }_0^{1}f(x)dx=0$ and determines $y$ only up to a constant.