I would like to understand, from an analytical point of view, why is it that we only take the first and second derivatives into account when computing the generator of a diffusion. This question is based on the following argument, to be found in Petr Mandl's book: Analytical treatment of one dimensional Markov processes page 12
In the proof of equation (20) the author expands only until the second order and then used the continuity of $f''$ Why didn't he go further and considered $f'''(\zeta(y))$. Or else, why didn't he stop before and just considered the first derivative, $f'(\zeta(y))$?