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In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open setΩ⊂ℝn{\Omega\subset\mathbb{R}^{n}}and givenp&gt;1{p&gt;1}, we study the blow-up of functionsu∈GSBV⁢(Ω){u\in\mathrm{GSBV}(\Omega)}, whose jump sets belong to an appropriate classp{\mathcal{J}_{p}}and whose approximate gradients are  p -th power summable. In analogy with the theory of  p -capacity in the context of Sobolev spaces, we prove that the blow-up of  u  converges up to a set of Hausdorff dimension less than or equal ton-p{n-p}. Moreover, we are able to prove the following result which in the case ofW1,p⁢(Ω){W^{1,p}(\Omega)}functions can be stated as follows: wheneveruk{u_{k}}strongly converges to  u , then, up to subsequences,uk{u_{k}}pointwise converges to  u  except on a set whose Hausdorff dimension is at mostn-p{n-p}.

On the blow-up of GSBV functions under suitable geometric properties of the jump set