Talk:Tanaka equation

My question: The sketch of the proof tells me to apply Ito's lemma to each approximation of the absolute value function |.| by a splice between a parabola and the absolute value function itself, where the splicing points are -\epsilon and \epsilon. Such a spliced function is not C^2. Does this not create a problem when applying Ito's lemma? Textbooks that I saw use C^{\infty} approximations around zero. Thanks for clearing this up for me.