Metric Tietze two-thirds approximation ($\Vert u{\Vert }_{\infty }\le M/3,\Vert u{|}_Y-g{\Vert }_{\infty }\le 2M/3$)

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.