In the book Statistical Learning with Sparsity: The Lasso and Generalizations, in section 2.4.1, they mention that the absolute value of $\beta$ has no derivative at $\beta=0$, therefore they proceed by direct inspection to determine the value of $\beta$ in a piece-wise manner, which is shown in 2.10. Does anyone know what they mean by direct inspection? Did they draw the function out and inspect it? I can't see how they arrived at 2.10 from inspecting 2.9.
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I can't help with the notion of inspection.. but I have given a full derivation with all of the steps and assumptions here: https://stats.stackexchange.com/questions/123672/coordinate-descent-soft-thresholding-update-operator-for-lasso/351134#351134 -> hope it helps – Xavier Bourret Sicotte Oct 22 '19 at 18:14