Approximate dynamical systems on interval

Let II be a an open real interval. We show that if a function H:I×R→IH:I×R→I satisfies the inequality |H(H(x0,s),t)−H(x0,s+t)|≤δfor s,t∈R with a δ≥0δ≥0 and an x0∈Ix0∈I such that the function H(x0,⋅)H(x0,⋅) is a continuous surjection of RR onto II, then there exists a dynamical system FF on II such that |H(x,t)−F(x,t)|≤9δfor x∈I,t∈R.