Individuals are constantly processing external information and translating it into actions. This draws on limited resources of attention and requires economizing on attention devoted to signals related to economic behaviour. A natural measure of such costs is based on Shannon’s “channel capacity”. Modelling economic agents as constrained by Shannon capacity as they process freely available information turns out to imply that discretely distributed actions, and thus actions that persist across repetitions of the same decision problem, are very likely to emerge in settings that without information costs would imply continuously distributed behaviour. We show how these results apply to the behaviour of an investor choosing portfolio allocations, as well as to some mathematically simpler “tracking” problems that illustrate the mechanism. Trying to use costs of adjustment to explain “stickiness” of actions when interpreting the behaviour in our economic examples would lead to mistaken conclusions.