Rademacher's theorem

Remarks

For $n = 1$ the theorem follows from Lebesgue's theorem: a Lipschitz function of one variable is absolutely continuous, hence of bounded variation, hence differentiable almost everywhere. In general one works with each coordinate of $f$, so one may take $m = 1$. Applying the case $n = 1$ on lines and Fubini's theorem, the directional derivative of $f$ in each fixed direction exists almost everywhere. One then shows that almost everywhere it depends linearly on the direction, first for a countable dense set of directions, and the Lipschitz bound turns this into full differentiability. For example, $f(x) = \norm{x}$ is Lipschitz on $\R^n$ and differentiable everywhere except at $0$. Continuity alone is not enough: the Weierstrass function is continuous on $\R$ and differentiable nowhere.

Going deeper

For a student

The theorem is the starting point of analysis on rough sets and maps. It lets one use the derivative of a Lipschitz map in integrals, for example in the area and coarea formulas of geometric measure theory. It also shows that Lipschitz functions belong to the Sobolev space $W^{1,\infty}$ locally, with weak gradient equal to the almost everywhere gradient. The one-dimensional case is closely tied to absolute continuity and to Lebesgue's theorem on monotone functions.