View source for Rademacher theorem

{{MSC|26B05|26B35}}

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[[Category:Analysis]]

A theorem proved by H. Rademacher about the differentiability of [[Lipschitz Function|Lipschitz functions]] .

'''Theorem'''
Let $U$ be an open subset of $\mathbb R^n$ and $f:U\to \mathbb R^k$ a Lipschitz function, i.e. such that there is a constant $C$ with 
\[
|f(x)-f(y)|\leq C|x-y|\qquad \mbox{for every } x,y\in U\, .
\]
Then $f$ is differentiable almost everywhere (with respect to the Lebesgue measure $\lambda$). That is, there is a set $E\subset U$ with $\lambda (U\setminus E) = 0$ and such that for every $x\in E$ there is a linear function $L_x:\mathbb R^n\to \mathbb R^k$ with
\[
\lim_{y\to x} \frac{f (x) - f(y) - L_x (y-x)}{|y-x|} \; =\; 0\, .
\]

For a proof see Theorem 2.14 of {{Cite|AFP}} or {{Cite|EG}}.

The same conclusion holds also for maps in the Sobolev class $W^{1,p}$ if $p$ is strictly larger then the dimension of the domain. A closely related, and more general, result is [[Stepanov theorem]] and
indeed some author use the terminology "Rademacher-Stepanov theorem".

===References===
{|
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|valign="top"|{{Ref|AFP}}||     L. Ambrosio, N. Fusco, D.  Pallara, "Functions of bounded variations    and  free discontinuity  problems". Oxford Mathematical Monographs.  The    Clarendon Press,  Oxford University Press, New York, 2000.      {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|}