Rigidity of the chain rule and nearly submultiplicative functions

Assume that T : C¹→(ℝ) C(ℝ) nearly satisfies the chain rule in the sense that (Formula presented) holds for all f, g ∈ C¹(ℝ) and x ∈ ℝ, where S: ℝ³ → ℝ is a suitable fixed function. We show under a weak non-degeneracy and a weak continuity assumption on T that S may be chosen to be 0, i.e. that T satisfies the chain rule operator equation, the solutions of which are explicitly known. We also determine the solutions of one-sided chain rule inequalities like (Formula presented) under a further localization assumption. To prove the above results, we investigate the solutions of nearly submultiplicative inequalities on ℝ (Formula presented) and characterize the nearly multiplicative functions on ℝ (Formula presented) under weak restrictions on φ.