Simple examples of non-Hermitian Hamiltonians with purely real spectra defined in L-2(R+) having spectral singularities inside the continuous spectrum are given. It is shown that such Hamiltonians may appear by shifting the independent variable of a real potential into the complex plane. Also they may be created as SUSY partners of Hermitian Hamiltonians. In the latter case spectral singularities of a non-Hermitian Hamiltonian are ordinary points of the continuous spectrum for its Hermitian SUSY partner. Conditions for transformation functions are formulated when a complex potential with complex eigenenergies and spectral singularities has a SUSY partner with a real spectrum without spectral singularities. Finally, we shortly discuss why Hamiltonians with spectral singularities are 'bad'.
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