= Metric Tietze two-thirds approximation
{title2=$\|u\|_\infty\le M/3,\quad\|u|_Y-g\|_\infty\le2M/3$}
For a real function $g$ in the <bounded continuous functions> of norm $M$ on a closed subset $Y$ of a <metric space>, separate its closed level sets $A=\{g\le-M/3\}$ and $B=\{g\ge M/3\}$ by $u=(M/3)(d(\cdot,A)-d(\cdot,B))/(d(\cdot,A)+d(\cdot,B))$. The <distance to a set> functions make $u$ continuous with the displayed bounds. Empty level sets are handled by the constants $M/3$, $-M/3$ or $0$. Iterating on the residual yields a uniformly convergent continuous extension with no increase in the <supremum norm>.
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