Consider an SDE for an Ito process $X_t$ with drift and diffusion $(\mu_t, \sigma_t)$ which exhibits a strong solution. Consider also a function $f(x)$ which exhibits a simple kink singularity at some point $c$. For example $$ f(x) = \begin{cases} ax \hspace{53pt} \text{ if } x\leq c \\ ac + b(x - c) \hspace{7pt} \text{ otherwise} \end{cases} $$ where $a \neq b$.
How would one apply Ito's lemma the process $Y_t = f(X_t)$?
Over the the majority of the domain, the standard version of Ito's lemma holds, but at the singular point the function is not differntiab