Approximate dynamical systems on interval

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