  I present a unified formulation of anomalous contributions in quantum field
theories by calculating directly the effective action using the background
field and covariant-derivative expansion technique. I use this method to
determine uniquely the induced Chern-Simons term from the a Lorentz and CPT
violating term in fermion QED Lagrangian. The outstanding ambiguity is resolved
by properly taken into account the noncommutivity of $A$ and $\partial$. The
resulting vacuum polarization tensor acquires a non-Feynman diagram anomalous
contribution which accounts for the discrepancy between the present calculation
and the other calculations.
