In this paper, we show that a process modeled by a strongly continuous real-valued semigroup (that has a space convolution operator as infinitesimal generator) cannot satisfy causality. We present and analyze a causal model of diffusion that satisfies the semigroup property at a discrete set of time points and that is in contrast to the classical diffusion model not smooth. More precisely, if v denotes the concentration of a substance diffusing with constant speed, then v is continuous, but its time derivative is discontinuous at the discrete set M of time points. Furthermore, we show that diffusion with constant speed satisfies an inhomogeneous wave equation with a time dependent coefficient. Copyright © 2011 John Wiley & Sons, Ltd.