I'm aware there is a very relevant explanation on L1 regularization's effect on feature selection at here: Why L1 norm for sparse models [Ref. 1].
To better understand it I'm reading Google's tutorial on Regularization for Sparsity: L₁ Regularization [Ref. 2]. When it comes to the following part, there's some statements I emphasized that I do not understand:
You can think of the derivative of L1 as a force that subtracts some constant from the weight every time. However, thanks to absolute values, L1 has a discontinuity at 0, which causes subtraction results that cross 0 to become zeroed out. For example, if subtraction would have forced a weight from +0.1 to -0.2, L1 will set the weight to exactly 0. Eureka, L1 zeroed out the weight.
I imagine when it says "L1 has a discontinuity at 0" it means the loss of the L1 like in the following figure [Ref. 1]:
But why it will "cause subtraction results that cross 0 to become zeroed out"? Why "if subtraction would have forced a weight from +0.1 to -0.2, L1 will set the weight to exactly 0"?
Is it related to that L1 is not differentiable at $w = 0$?