We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Maly [Ma1] to the Sobolev spaces on metric spaces [H1].
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Piotr Hajłasz, Juha Kinnunen, Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoam. 14 (1998), no. 3, pp. 601–622DOI 10.4171/RMI/246