  Paradoxes in the Boltzmann kinetic theory are presented. Firstly, it is
pointed out that the usual notion concerning the perfect continuity of
distribution function is not generally valid; in many important situations
using certain types of discontinuous distribution functions is an absolute
must. Secondly, it is revealed that there is no time reversibility in terms of
beam-to-beam collisions and, in connection with this, there are intrinsic
difficulties in formulating the net change of molecular density due to
collisions, either in the three-dimensional velocity space or in the
six-dimensional phase space. With help of simple examples, the paradoxes
manifest themselves clearly.
