Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces

We show that on complete doubling metric measure spaces X supporting a Poincare inequality, all Newton-Sobolev functions u are quasicontinuous, i.e. that for every a>0 there is an open subset U of X with capacity less than a and such that the restriction of u to X\U is continuous. This implies that the capacity is an outer capacity