Let \(C_m (I)\) denote the space of \(m\) times continuously differentiable functions on a real interval \(I\), and let \(P_n (I)\) be the subset of \(C_n (I)\) consisting of all polynomials of degree \(\leq n\). The authors prove the following theorem: if the superposition operator \(Fu (x)= f(x,u(x))\) generated by some function \(f: I\times \mathbb{R}\to \mathbb{R}\) is globally Lipschitz from \(P_n (I)\) into \(C_m (I)\), then \(f\) is affine in the last argument.