In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term f (u) = λ|u| α u. We show that low regularity of f (i.e., α > 0 but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE wt = f (w). This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for NLS in certain H s spaces, which depend on the smallness of α rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel's formula. This yields in particular that if α is sufficiently small and N sufficiently large, then the nonlinear heat equation is ill-posed in H s (R N) for all s ≥ 0.