Necessary and sufficient conditions for the invertibility of the non-linear difference operator $(\mathscr Dx)(t)=x(t+1)-f(x(t))$ in the space of bounded continuous functions on the real axis

Necessary and sufficient conditions for the invertibility in the space of bounded continuous functions on $\mathbb R$ of the non-linear difference operator
$$ (\mathscr Dx)(t)=x(t+1)-f(x(t)), \qquad t\in\mathbb R, $$
with $f\colon\mathbb R\to\mathbb R$ a continuous map, are obtained.