Stepanov's theorem

Stepanov's theorem is a generalization of Rademacher 's theorem on the differentiability of a Lipschitz function.

Suppose function $f$ ${\ displaystyle f}$ $f$ defined on an open set $\Omega$ ${\ displaystyle \ Omega}$ $\ Omega$ Euclidean space $A\subset \Omega$ ${\ displaystyle A \ subset \ Omega}$ $A \ subset \ Omega$ and

\varlimsup _{x\to a}{\frac {|f(x)-f(a)|}{|x-a|}}<\infty

{\ displaystyle \ varlimsup _ {x \ to a} {\ frac {| f (x) -f (a) |} {| xa |}} <\ infty}

{\ displaystyle \ varlimsup _ {x \ to a} {\ frac {| f (x) -f (a) |} {| x-a |}} <\ infty}

for all $a\in A$ ${\ displaystyle a \ in A}$ $a \ in A$ . Then $f$ ${\ displaystyle f}$ $f$ differentiable almost everywhere in $A$ ${\ displaystyle A}$ $A$ .

Proved by Stepanov ^[1] .

Literature

Federer G., Geometric theory of measure, 1987, p. 236, (Theorem 3.1.9)

Notes

↑ H. Stepanoff: Über totale Differenzierbaгkeit. Math. Ann. 90 (1923), 318-320.

[1] H. Stepanoff: Über totale Differenzierbaгkeit. Math. Ann. 90 (1923), 318-320.

Stepanov's theorem

Literature

Notes

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