A flat function is a smooth function all of whose derivatives vanish at a given point $x_0$. Why can such a function be anything but a constant function? Intuitively if a function's derivative and the derivative of the derivative, ... are zero then some tiny step to the right the function hasn't changed, the dirivitive hasn't changed, ... so we are in the same situation again: It seems the function should be constant. Yet clearly, there are examples of non-constant flat functions!
E.g. If a particle has no velocity and also no acceleration and also the derivative of the acceleration, ... , are all zero, the particle shouldn't move at all!
This is a duplicate of this question which received no helpful answer. Thanks!