The problem of a fermion subject to a convenient mixing of vector and scalar potentials in a two-dimensional space-time is mapped into a Sturm-Liouville problem. For a specific case which gives rise to an exactly solvable effective modified Pöschl-Teller potential in the Sturm-Liouville problem, bound-state solutions are found. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail and some unusual results are revealed. The Dirac delta potential as a limit of the modified Pöschl-Teller potential is also discussed. The problem is also shown to be mapped into that of massless fermions subject to classical topological scalar and pseudoscalar potentials. Copyright © EPLA, 2007.