Why more smooth the function the more precise finite difference method? As the title, Why more smooth the function the better finite difference method? 
I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? 
Thanks

You already said it: The consistence (local truncation error) of finite difference schemes is derived using Taylor expansions. Taylor's theorem shows that you can expand a function depending on how smooth it is. For smooth functions, this allows you to construct larger stencils with higher accuracy, since the higher derivatives are bounded. This is why finite differences can be used without any harm if you know in advance your solution fulfills some regularity. However, this is seldom the case for real-world problems. This led to the development of finite Element/Volume methods.
