  The continuity conditions of the radial distribution function g(r) and its
close relative the cavity function y(r) are studied in the context of the
Percus-Yevick (PY) integral equation for 3D square-well fluids. The cases
corresponding to a well width, (w-1)*d, equal to a fraction of the diameter of
the hard core, d/m, with m=1,2,3 have been considered. In these cases, it is
proved that the function y(r) and its first derivative are everywhere
continuous but eventually the derivative of some order becomes discontinuous at
the points (n+1)d/m, n=0,1,... . The order of continuity (the highest order
derivative of y(r) being continuous at a given point) is found to be
proportional to n in the first case (m=1) and to 2*n in the other two cases
(m=2,3), for large values of n. Moreover, derivatives of y(r) up to third order
are continuous at r=d and r=w*d for w=3/2 and w=4/3 but only the first
derivative is continuous for w=2. This can be understood as a non-linear
resonance effect.
