Let $g$ be a continuous function (not necessarily smooth). Let $\Delta g=\frac{g(x+\Delta x)-g(x)}{\Delta x}$.
Consider the difference equation:
$F(\Delta\Delta g,\Delta g,g)=K$
Question: Under which assumptions, would the difference equation to have the same solutions as the ODE $F(f'',f',f)=K$?
"Assumption" means, for example, some additional restrictions putting on $\Delta x$ and $g$.