I was going through the derivation of soft threholding at http://dl.dropboxusercontent.com/u/22893361/papers/Soft%20Threshold%20Proof.pdf.
It says the three unique solutions for
$\operatorname{arg min} \|x-b\|_2^2 + \lambda\|x\|_1$ is given by
$\operatorname{arg min} \|x-b\|^2 + \lambda x$ assuming $x > 0$
$\operatorname{arg min} \|x-b\|^2 - \lambda x$ assuming $x < 0$
$\operatorname{arg min} \|x-b\|^2 = 0 $ assuming $x = 0$
I didn't get how it was derived. Any help?