  Cooper instability associated with superconductivity in the two-dimensional
semi-Dirac semimetals is attentively studied in the presence of attractive
Cooper-pairing interaction, which is the projection of an attractive
fermion-fermion interaction. Performing the standard renormalization group
analysis shows that the Cooper theorem is violated at zero chemical potential
but instead Cooper instability can be generated only if the absolute strength
of fermion-fermion coupling exceeds certain critical value and transfer
momentum is restricted to a confined region, which is determined by the initial
conditions. Rather, the Cooper theorem would be instantly restored once a
finite chemical potential is introduced and thus a chemical potential-tuned
phase transition is expected. Additionally, we briefly examine the effects of
impurity scatterings on the Cooper instability at zero chemical potential,
which in principle are harmful to Cooper instability although they can enhance
the density of states of systems. Furthermore, the influence of competition
between a finite chemical potential and impurities upon the Cooper instability
is also simply investigated. These results are expected to provide instructive
clues for exploring unconventional superconductors in the kinds of semimetals.
