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A subset of ${ℝ}^d$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f:{ℝ}^d\to ℝ$. 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.