Confusion related to the calculation of gradient

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I am having some confusion related to the calculation of gradient.

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My function $f(X) = g(X) + \lambda||X||_1$ where g(X) is convex and differentiable. I didn't get how the second expression when $X_{ij}=0$ is derived

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Well I found the reason. It's related to minimum norm subgradient http://www.di.ens.fr/~mschmidt/Documents/2011_OPT_Chapter.pdf.