Green's function on the unit interval $[0,1]$ I would like to find the Green's function on the unit interval $[0,1]$. We have to solve the linear equation $$\frac{d^2f(x)}{dx}=g(x)$$ with boundary condition $f(0)=a$, $f(1)=b$. So the Green's function satisfies $$\frac{d^2G(x,x')}{dx}=\delta(x-x'),$$ where $δ$ is the Dirac delta function. Essentially, $$G(x,x') = A_1x+A_2$$ with $0 \le x \le x'$ and $$G(x,x')= A_3x+A_4$$ with $x' \le x \le 1$. At this point, I'd to find the constants $A_1$, $A_2$, $A_3$ and $A_4$. We know the boundary conditions : $$G(0,x')=A_1 \cdot 0 + A_2=a \implies A_2=2$$ $$G(1,x')=A_3+A_4=b.$$ Furthermore, $$\int_{x-\epsilon}^{x+\epsilon} \frac{d^2G}{dx^2}dx = \frac{dG}{dx}\big|_{x+\epsilon}+\frac{dG}{dx}\big|_{x-\epsilon}=\int_{x-\epsilon}^{x+\epsilon} \delta(x-x')=1 \implies A_3-A_1=1$$
Here, I have three equations for four constants. Questions : How could I find another equation to find the "last" constant? Should we involve the continuity of $G$?
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You are missing the Green's function continuity at $x = x'$. See below.

Write your solution as
$$
{\rm f}\pars{x} =
a + \pars{b - a}x + \int_{0}^{1}{\rm G}\pars{x,x'}{\rm g}\pars{x'}\,\dd x'
$$
such that your Green's function satisfies homogeneous boundary conditions:
$$
{\rm G}\pars{0,x'} = {\rm G}\pars{1,x'} = 0\,\qquad\forall\ x' \in \pars{0,1}
$$
Then,
$$
{\rm G}\pars{x,x'} =
\left\lbrace\begin{array}{lcrcl}
Ax & \mbox{if} & x & < & x'
\\
B\pars{1 - x} & \mbox{if} & x & > & x'
\end{array}\right.
$$
Also,
$$
\left.\partiald{{\rm G}\pars{x,x'}}{x}\right\vert_{x\ =\ x'^{-}}^{x\ =\ x'^{+}} = 1
\qquad\mbox{and}\qquad{\rm G}\pars{x,x'^{-}} = {\rm G}\pars{x,x'^{+}}
$$
yield
$$
\left.\begin{array}{rcrcl}
A & + & B & = & - 1
\\
x'A & + & \pars{x' - 1}B & = & 0
\end{array}\right\rbrace
\qquad\imp\qquad
\left\lbrace\begin{array}{rcl}
A & = & \phantom{-}x' - 1
\\
B & = & -x'
\end{array}\right.
$$
$$
\begin{array}{|c|}\hline\mbox{}\\
{\rm G}\pars{x,x'} =
\left\lbrace\begin{array}{lcrcl}
\pars{x' - 1}x & \mbox{if} & x & < & x'
\\
\pars{x - 1}x' & \mbox{if} & x & > & x'
\end{array}\right.
\\ \mbox{}\\ \hline
\end{array}
$$
