We study the one-dimensional Dirac equation in the framework of a position
dependent mass under the action of a Woods-Saxon external potential. We find
that constraining appropriately the mass function it is possible to obtain a
solution of the problem in terms of the hypergeometric function. The mass
function for which this turns out to be possible is continuous. In particular
we study the scattering problem and derive exact expressions for the reflection
and transmission coefficients which are compared to those of the constant mass
case. For the very same mass function the bound state problem is also solved,
providing a transcendental equation for the energy eigenvalues which is solved
numerically.Comment: Version to match the one which has been accepted for publication by
J. Phys. A: Math. Theor. Added one figure, several comments and few
references. (24 pages and 7 figures
