# Ito formula for non $C^{2}$ function

In the book Brownian motion and Stochastic Calculus by Karatzas and Shreve, page 215, there is a problem that says the following:

Let $$a_{1}, ..., a_{n}$$ be real numbers and denote $$D=\lbrace a_{1}, ..., a_{n}\rbrace$$. Suppose that $$f: \mathbb{R}\rightarrow \mathbb{R}$$ is continuous and $$f'$$ and $$f''$$ exist and are continuous on $$R/D$$ and the lateral limits of the derivatives at the points in $$D$$ exist and are finite. Show that $$f$$ is the difference of two convex functions and,

$$f(W_{t}) = f(z) + \int_{0}^{t}f'(W_{s})dW_{s} + \frac{1}{2}\int_{0}^{t}f''(W_{s})ds$$

$$+ \sum_{k=1}^{n}L_{t}(a_{k})(f'(a_{k}+) - f'(a_{k}-))$$

where $$L_{t}(a) = |W_{t} - a| - |a| - \int_{0}^{t}sign(W_{s}-a)dW_{s}$$

My questions is, does this version of Ito's formula holds for any Ito process, i.e

$$f(X_{t}) = f(z) + \int_{0}^{t}f'(X_{s})dX_{s} + \frac{1}{2}\int_{0}^{t}f''(X_{s})ds$$

$$+ \sum_{k=1}^{n}L_{t}(a_{k})(f'(a_{k}+) - f'(a_{k}-))$$

where $$L_{t}(a) = |X_{t} - a| - |X_{0} - a| - \int_{0}^{t}sign(X_{s}-a)dX_{s}$$

If this is true, in the case where $$f'$$ is continuous and only $$f''$$ is discontinuous at a finite number of points, this equation reduces to the 'regular' Ito formula. Correct?