# Article

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Keywords:
differentiability; Lipschitz functions; universal differentiability set; $\sigma$-porous set
Summary:
A subset of $\mathbb R^{d}$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f\colon\mathbb R^{d}\to \mathbb R$. We show that any universal differentiability set contains a `kernel' in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.
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