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The first moment of the chirality-odd twist-3 parton distribution e(x) is related to the pion-nucleon σ-term, which is important for phenomenology. However, the possible existence of a singular contribution proportional to δ(x) in the distribution prevents the determination of the σ-term with e(x) extracted from experimental data. There are two approaches to show the existence: the first one is based on an operator identity; the second one is based on a perturbative calculation of a single quark state with finite quark mass. We show that all contributions proportional to δ(x) in the first approach are canceled. For the second approach we find that e(x) of a multiparton state with a massless quark has no contribution with δ(x). Considering that a proton is essentially a multiparton state, the effect of the contribution with δ(x) is expected to be suppressed by light quark masses with arguments from perturbation theory. A detailed discussion of the difference between cut diagrams and uncut diagrams of e(x) is provided.