= Solution
A physical bound-state <wave function> must be single-valued, finite, square-integrable and normalizable. Across a finite step in a regular potential, both $\psi$ and its first derivative are continuous when the particle <mass> is the same on either side; integrate the <Schrödinger equation> across an infinitesimal interval to obtain the derivative condition. The <second derivative> need not be continuous.
\b[An infinite hard wall is an exception to derivative matching]: the interior <wave function> vanishes at the boundary, but its derivative need not equal the exterior zero derivative. A delta-function potential also permits a prescribed derivative jump, and an effective-mass interface requires the appropriate mass-weighted <matching condition>. Thus ordinary finite-step conditions should not be asserted without qualification for every singular potential.
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