*Preprint*

**Inserted:** 15 dec 2016

**Last Updated:** 15 dec 2016

**Year:** 2016

**Abstract:**

In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $\mu$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $\mu$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(\mu,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point $x$ if it is null at the origin and it is the sum of a linear function on $V(\mu,x)$ and a Lipschitz function on $V(\mu,x)^{\perp}$.

**Keywords:**
Lusin type approximation, Lipschitz function, Radon measure, Blowup

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