In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Sturm-Liouville equation
$$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\epsilon \phi(x)$$
where
$$m(x) = \left\{\begin{matrix}m^-, x<0 \\ m^+, x>0\end{matrix}\right.$$
is a step type function, and defines two separate regions in which we can solve for the wavefunction
$$\phi(x) = \left\{\begin{matrix}\psi^-(x), x<0 \\ \psi^+(x), x>0\end{matrix}\right.$$
with the help of boundary conditions. One of the boundary conditions that is used at the discontinuity in $m(x)$ is that $\phi(x)$ is continuous
$$\psi^-(0)=\psi^+(0)$$
What is the justification/derivation for this condition? Normally I would assume it is a physical assumption, but I don't think this is a good enough reason as later in the text the author studies a similar problem for the Dirac equation, where the wavefunction turns out to be discontinuous at the "step". On the wikipedia page for the S-L equation it states that the solution $\phi(x)$ should be continuously differentiable, but this is for the simple case when the coefficients are continuous. I do not know how the situation changes in this example.
All types of answers are gratefully received, but I am a physicist so be gentle if possible ;)
From the point of view of Mathematics, the following equation can be solved on an interval $[a,b]$ with $c\in[a,b]$, for all choices of constants $A$, $B$: $$ -\frac{d}{dx}\left(p(x)\frac{df}{dx}\right) +q(x)f(x) = \lambda w(x) f(x),\\ a \le x \le b, \;\;\; f(c) = A,\; f'(c)=B. $$ You can get by with very few assumptions on the coefficients if you want reformulate the problem as an integral equation. For example, suppose $p$, $q$, $1/p$ are Lebesgue integrable on $[a,b]$, and suppose that $p$ is continuous at $c$. Then you can reformulate the above as an integral equation
From a Mathematical point of view, the integral equation is the correct theoretical description of the equation--it has a unique solution given any $A$, $B$, at least under the conditions stated above. You can show that a full spectral theory can be developed using solutions of the integral equation, which makes the integral equation cleanly tied to solvability. There is no natural cause or natural way to split $[a,b]$ into two intervals $[a,c]$, $[c,b]$ because the solutions of the integral equation extend uniquely to the entire interval.
It is always possible to solve separate problems on $[a,c]$ and on $[c,b]$, and then to specify conditions for tying the two solutions together $c$; but that's not the same as solving the problem on $[a,b]$, where there are no conditions at $c$ other than natural continuity of the integral equation solution and its first derivative across $c$. Allowing a discontinuity at $c$ where $p(c)$ is discontinuous is natural, but one must determine coupling rules between the problems on $[a,c]$ and $[c,d]$, which requires (or defines) some Physical cause at $c$.