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For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$,the properties are studied of fields of `good directions', that is thedirections with respect to which $\partial\Omega$ can be locally represented asthe graph of a continuous function. For any such domain there is a canonicalsmooth field of good directions defined in a suitable neighbourhood of$\partial\Omega$, in terms of which a corresponding flow can be defined. Usingthis flow it is shown that $\Omega$ can be approximated from the inside and theoutside by diffeomorphic domains of class $C^\infty$. Whether or not the imageof a general continuous field of good directions (pseudonormals) defined on$\partial\Omega$ is the whole of $\mathbb{S}^{m-1}$ is shown to depend on thetopology of $\Omega$. These considerations are used to prove that if $m=2,3$,or if $\Omega$ has nonzero Euler characteristic, there is a point$P\in\partial\Omega$ in the neighbourhood of which $\partial\Omega$ isLipschitz. The results provide new information even for more regular domains,with Lipschitz or smooth boundaries.

Partial regularity and smooth topology-preserving approximations of rough domains