  Kolmogorov famously proved that multivariate continuous functions can be
represented as a superposition of a small number of univariate continuous
functions, $$ f(x_1,\dots,x_n) = \sum_{q=0}^{2n+1} \chi^q \left( \sum_{p=1}^n
\psi^{pq}(x_p) \right).$$ Fridman \cite{fridman} posed the best smoothness
bound for the functions $\psi^{pq}$, that such functions can be constructed to
be Lipschitz continuous with constant 1. Previous algorithms to describe these
inner functions have only been H\"older continuous, such as those proposed by
K\"oppen and Braun and Griebel. This is problematic, as pointed out by Griebel,
in that non-smooth functions have very high storage/evaluation complexity, and
this makes Kolmogorov's representation (KR) impractical using the standard
definition of the inner functions.
  To date, no one has presented a method to compute a Lipschitz continuous
inner function. In this paper, we revisit Kolmogorov's theorem along with
Fridman's result. We examine a simple Lipschitz function which appear to
satisfy the necessary criteria for Kolmogorov's representation, but fails in
the limit. We then present a full solution to the problem, including an
algorithm that computes such a Lipschitz function.
