We are given an unknown univariate Lipschitz continuous function that we wish to estimate by evaluating the function sequentially at distinct points. We provide a procedure for recursively selecting this sequence of points so that, averaging over points in the domain the resulting worst case error between the estimating and actual functions is minimized. Upper and lower bounds on these errors is also provided.Scope and purposeIn engineering and science, it is often necessary to estimate functions based on a small number of evaluations. In this paper we determine which points should be evaluated in order to maximize the information gained at each evaluation. In particular, we prove that the sampling strategy that minimizes worst case error, averaged over points in the domain, is to sample the midpoint of a specific interval. We provide an estimation procedure that bounds a function using a Lipschitz bracket. The resulting estimating function is simple and consists of linear functions only.