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Let $P$ be a finite simplicial comple with underlying space (union ofsimplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Thenthere exists $K < \infty$, \emph{depending only on $a$ and $Q$,} with thefollowing property. Let $\mathcal{S} \subset |P|$ be closed and suppose $\Phi$is a continuous map of $|P| \setminus \mathcal{S}$ into some topological space$\mathcal{F}$. Suppose $\dim (\tilde{\mathcal{S}} \cap |Q|) \leq a$, where"$\dim$" = Hausdorff dimension. Then there exists $\tilde{\mathcal{S}} \subset|P|$ such that $\tilde{\mathcal{S}} \cap |Q|$ is the underlying space of asubcomplex of $Q$ and there is a continuous map $\tilde{\Phi}$ of $|P|\setminus \tilde{\mathcal{S}}$ into $\mathcal{F}$ such that $\mathcal{H}^{a}\bigl(\tilde{\mathcal{S}} \cap |Q| \bigr) \leq K \mathcal{H}^{a}\bigl(\mathcal{S} \cap |Q| \bigr)$, where $\mathcal{H}^{a}$ denotes$a$-dimensional Hausdorff measure; if $x \in \tilde{\mathcal{S}}$ then $x$belongs to a simplex in $P$ intersecting $\mathcal{S}$; if $x \in |P| \setminus\mathcal{S}$, $x \in \sigma \in P$, and $\sigma$ does not intersect any simplexin $Q$ whose simplicial interior intersects $\mathcal{S}$, then$\tilde{\Phi}(x)$ is defined and equals $= \Phi(x)$; if $\sigma \in P$ then$\tilde{\Phi}(\sigma \setminus \tilde{\mathcal{S}}) \subset \Phi(\sigma\setminus \mathcal{S})$; and if $\mathcal{F}$ is a metric space and $\Phi$ islocally Lipschitz on $|P| \setminus \mathcal{S}$ then $\tilde{\Phi}$ is locallyLipschitz on $|P| \setminus \tilde{\mathcal{S}}$ Moreover, $P$ can be replacedby an arbitrarily fine subdivision without changing $K$.

On the Approximation of a Function Continuous off a Closed Set by One Continuous Off a Polyhedron