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A quasiregular (QR) map is a Sobolev map f: ℝⁿ → ℝⁿ satisfying the distortion inequality |Df(x)|ⁿ ≤ K det(Df(x)) at almost every x, where K ≥ 1 is a constant. QR maps form a higher-dimensional class of maps with many similar properties to holomorphic maps, such as continuity, openness, discreteness, and versions of the Liouville, Picard and Montel theorems. In this talk, we consider a generalization of the distortion inequality of the form |Df(x)|ⁿ ≤ K det(Df(x)) + Σ(x)|f(x) - y|, where Σ is a locally Lᵖ-integrable function for p > 1 and y ∈ ℝⁿ is a fixed point. We show that this condition yields counterparts to many fundamental properties of QR maps at the single point y, including single-value versions of openness, discreteness, and the Liouville theorem. Joint work with Jani Onninen.