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We study the one-dimensional Dirac equation in the framework of a positiondependent mass under the action of a Woods-Saxon external potential. We findthat constraining appropriately the mass function it is possible to obtain asolution of the problem in terms of the hypergeometric function. The massfunction for which this turns out to be possible is continuous. In particularwe study the scattering problem and derive exact expressions for the reflectionand transmission coefficients which are compared to those of the constant masscase. For the very same mass function the bound state problem is also solved,providing a transcendental equation for the energy eigenvalues which is solvednumerically.

New exact solution of the one dimensional Dirac Equation for the Woods-Saxon potential within the effective mass case