When sufficiently small perturbations of parameters preserve strict preference for one alternative over another, dependence on the parameters is continuous. We characterise this property with a utility function over alternatives that depends continuously on the parameter. The class of parameter spaces such that this form of representation is guaranteed to exist is also characterised. When the parameters are beliefs, these results have implications for robust portfolio choice, Bayesian games and psychological games. When alternatives are discrete, the representation is jointly continuous, and an extension of Berge’s theorem of the maximum, yields a continuous value function. We apply this result to generalise a standard consumer choice problem: where parameters are price-wealth vectors. When the parameter space is lexicographically ordered, a novel application to referencedependent preferences is possible.