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We show that capacity can be computed with locally Lipschitz functions inlocally complete and separable metric spaces. Further, we show that if$(X,d,\mu)$ is a locally complete and separable metric measure space, thencontinuous functions are dense in the Newtonian space $N^{1,p}(X)$. Here themeasure $\mu$ is Borel and is finite and positive on all metric balls. Inparticular, we don't assume properness of $X$, doubling of $\mu$ or anyPoincar\'e inequalities. These resolve, partially or fully, questions posed bya number of authors, including J. Heinonen, A. Bj\"orn and J. Bj\"orn. Incontrast to much of the past work, our results apply to locally complete spaces$X$ and dispenses with the frequently used regularity assumptions: doubling,properness, Poincar\'e inequality, Loewner property or quasiconvexity.

Density of continuous functions in Sobolev spaces with applications to capacity