A discontinuity in the solutions for the Schrodinger equation is usually not acceptable. Both the Schrodinger equation and certain operators (e.g. the momentum operator $\hat{p}$ given by $-\imath \hbar d /dx$) require the solutions to be differentiable at every point in order to give a valid result. The criteria for differentiability of a function at a point requires that the function be continuous at that point. Although requiring continuity of the solutions in order to maintain differentibaility, the nature of the infinite well mandates that this continuity occur only within the well itself. It will be shown that the lack of continuity at the boundary does not affect either the normalization of the solutions or the results obtained when a linear operator $\hat{Q}$ is applied to the solutions. Important to the analysis is the fact that the Time-Independent Schrodinger equation can be solved for potentials that can be separated into piecewise regions resulting in solutions that in turn can be combined to yield a composite solution throughout a larger region of interest. This is a standard method of analysis in solving simple systems.
A. Normalization.
With regards to normalization, a piecemeal potential can be normalized over all space in 1-dimension even if the function is discontinuous. In Figure 3, let $\Psi_{\text{I}}$, $\Psi_{\text{II}}$ and $\Psi_{\text{III}}$ represent the probability density functions in regions I, II and III respectively. Furthermore, let $\Psi_{\text{I}} =0$ in region I and $\Psi_{\text{III}} =0$ in region III. The entire probability density function can be normalized as follows
\[ \int_{-\infty}^{x_{0}} \Psi_{\text{I}}^{*} \cdot \Psi_{\text{I}} dx + \int_{x_{0}}^{x_{0}+L} \Psi^{*}_{\text{II}}(x)\Psi_{\text{II}}(x)dx + \int_{x_{0}+L}^{\infty} \Psi_{\text{III}}^{*} \cdot \Psi_{\text{III}} dx = 1 \]
Substituting $\Psi_{\text{I}}= 0 $ and $\Psi_{\text{III}}= 0 $, simply gives
\[ \int_{x_{0}}^{x_{0}+L} \Psi^{*}_{\text{II}}(x)\Psi_{\text{II}}(x)dx = 1 \]
B. The Domain of the Linear Operators.
With regards to the effect of a linear operator operating on the solutions, the regions outside the well can simply be considered as being outside the domain of the linear operator.
The expectation value $\langle q \rangle$ of an observable $q$ given by a linear operator $\hat{Q}$ is calculated over the entire 1-dimensional space for the infinite well as follows
\[ \langle q \rangle = \langle \Psi_{\text{I}} \Big | \hat {Q} \Psi_{\text{I}} \rangle + \langle \Psi_{\text{II}} \Big | \hat {Q} \Psi_{\text{II}} \rangle + \langle \Psi_{\text{III}} \Big | \hat {Q} \Psi_{\text{III}} \rangle \notag \]
\[ = \int_{- \infty}^{x_{0}} \Psi_{\text{I}}^{*} \hat{Q} \Psi_{\text{I}} dx + \int_{x_{0}}^{x_{0}+L} \Psi_{\text{II}}^{*} \hat{Q} \Psi_{\text{II}} dx+ \int_{x_{0}+L}^{\infty} \Psi_{\text{III}}^{*} \hat{Q} \Psi_{\text{III}} dx \tag{1} \label{Eq:UndefinedObservable} \]
However, the first and third terms on the right hand side of Eq.\eqref{Eq:UndefinedObservable} are equal to zero. Therefore, Eq.\eqref{Eq:UndefinedObservable} reduces to
\[ \langle q \rangle = \int_{x_{0}}^{x_{0}+L} \Psi_{\text{II}}^{*} \hat{Q} \Psi_{\text{II}} dx \tag{2} \label{Eq:Q} \]
Examination of Eq.\eqref{Eq:Q} shows that the linear operator effectively only is applicable to within the well and therefore regions in the exterior are irrelevant. Therefore, there is no need for continuity at the boundary.
A more practical example exploring the issue is as follows: The probability of a child being born into given population of women and men can be calculated. Furthermore, the expected hair color of the newborn can be determined. Similarly, it can be stated with certainty that the probability (i.e. the probability density function) of a child being born into a given population composed solely of men is zero. However, the question of what is the expected hair color (i.e. observable) of such a newborn is outside the scope (i.e. the domain) of a meaningful question.