= Solution
Away from the source point, the <Green function> obeys the homogeneous equation and must satisfy the appropriate endpoint condition. Thus it has the form $A\sin(kx)$ to the left of $\xi$ and $B\sin(k(\pi-x))$ to the right. It must be continuous at $\xi$: otherwise its second distributional derivative would contain an unwanted derivative of a <Dirac delta>. Integrating across $\xi$ gives the jump condition $G_x(\xi^+,\xi)-G_x(\xi^-,\xi)=1$.
The left and right homogeneous solutions $u=\sin(kx)$, $v=\sin(k(\pi-x))$ have constant <Wronskian>
$$
uv'-u'v=-k\sin(k\pi).
$$
This is nonzero for real noninteger $k$. Dividing the product of the left and right solutions by that Wronskian therefore gives the <Dirichlet Helmholtz Green function on an interval>
$$
\boxed{G(x,\xi)=-\frac{\sin(k\min(x,\xi))\sin(k(\pi-\max(x,\xi)))}{k\sin(k\pi)}.}
$$
The formula vanishes at both endpoints, is continuous at the source and has derivative jump one, verifying all distributional and boundary conditions. A difference of two such Green functions would be a homogeneous Dirichlet solution; it vanishes because $\sin(k\pi)\ne0$, proving uniqueness. Although zero is excluded by the stated noninteger condition, the removable $k\to0$ limit is $-\min(x,\xi)(\pi-\max(x,\xi))/\pi$.
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