Continuity of the fixed-point assignment ($d(\tau (f),\tau (g))\le d_{\infty }(f,g)/(1-q_f)$)

For contraction mappings $f,g$, compare their fixed points using $f$ on both arguments, then bound the change from $f$ to $g$ by the uniform metric. The resulting inequality gives the displayed estimate. Continuity at $f$ only needs its own contraction constant $q_f<1$, even if nearby maps have constants approaching one.