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In a Hermitian system, bound states must have quantized energies, whereas extended states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call ``continuum Landau modes'' (CLMs), have gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally-realizable 1D and 2D lattice models that can be used to study CLMs; the lattice eigenstates are localized and have other features that are the same as in the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires only a simple lattice design without nonreciprocal couplings.