This thesis presents the first known method to compute Lipschitz continuous inner functions for the Kolmogorov Superposition Theorem. While the inner functions of these superpositions can be Lipschitz continuous, previous algorithms that compute inner functions yield only Hölder continuous functions. These Hölder continuous functions suffer from high storage-to-evaluation complexity, thereby rendering the Kolmogorov Superposition Theorem impractical for computation. Once this concern is addressed, the Kolmogorov Superposition Theorem becomes an effective tool in dimension reduction to represent multivariate functions as univariate expressions, with applications in encryption, video compression, and image analysis. In this thesis, I posit sufficient criteria to iteratively construct a Lipschitz continuous inner function. I demonstrate that implementations of the predominant approach for such a Lipschitz construction do not satisfy these conditions. Instead, I manipulate Hölder continuous inner functions to create Lipschitz continuous reparameterizations. I show these reparameterizations meet my sufficient conditions, thereby guaranteeing that these reparameterized functions are inner functions for the Kolmogorov Superposition Theorem.