Surjectivity of a continuous bounded-displacement map
= Surjectivity of a continuous bounded-displacement map
If a <continuous map> $g:\mathbb R^n\to\mathbb R^n$ has $\|g(x)-x\|\leq K$, then it is onto. For a target $y$, choose $R\geq K+\|y\|$, $R>0$, and map the <unit ball> to itself by $f(x)=x-(g(Rx)-y)/R$. A <fixed point> gives $g(Rx)=y$.