Rademacher's Theorem (that every Lipschitz function on $\mathbb{R}^{n}$ is almost everywhere differentiable) is a remarkable result on the structure of the space of Lipschitz functions, but I was wondering whether it has any interesting applications. All of the "useful" results (or maybe "applicable") that I know of about weak versions of differentiability involve estimates (e.g. Sobolev embedding, Lebesgue differentiation theorem).