In this paper we introduce an algorithm for computing nonlinear continuous Chebyshev approximations. The algorithm is based on successive linearizations within adaptively adjusted neighborhoods. The convergence of the algorithm is proven under some general assumptions such that it is applicable for many Chebyshev approximation problems discussed in the literature. It, like the Remez exchange method, is purely continuous in the sense that it converges to a solution of a continuous Chebyshev

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... ximation problem rather than one on a discretized set. Quadratic convergence is shown in so-called regular cases, including polynomial and nondegenerate rational approximations. We believe the algorithm is also computationally more efficient than some other algorithms. A few numerial examples are given to illustrate the basic features of the algorithm.