stability of one dimensional dynamical system

Question

Suppose we have an iterated map $g(x)=x+f(x)$ where $f$ is a flat function at the origin (i.e. smooth, with all derivatives vanishing at zero). The standard stability test (for the origin) of looking at higher derivatives (note that $g'(0)=1$, so one needs to look into higher order derivatives till something non-vanishing pops up, and then depending on whether the order is even or odd and whether the derivative is positive or negative, one gets information about stability, instability or semi-stability as usual business) is non-conclusive here. Are there other methods to determine the dynamics of this flat perturbation of the identity map? Is it going to depend on the specific form of $f$? My guess is yes, and depends on whether it lowers or lifts the graph of the identity below or above the diagonal. Looking just at the right neighbourhood of $0$ for simplicity, if $f=e^{-\frac{1}{x^2}}$ we should get unstable as 0 seems repelling from a cobweb diagram point of view, if $f=-e^{-\frac{1}{x^2}}$ we should get asymptotic stability for similar reasons. This reasoning is very informal of course, but does it go in the right direction?