For a given function $g(t)=f(x(t))$. Assume $f()$ is smooth and suppose we know a constant $c \in \mathbb{R}$ such that $$\lim_{g(t)\to c^{-}} \frac{d}{dt}g(t)>0 \quad \text{and} \ \ \ \ \lim_{g(t)\to c^{+}} \frac{d}{dt}g(t)<0.$$ Do the limits around $c$ hold if we add a disturbance $d(x(t))$ to $g(t),$ i.e. $$\hat{g}(t)=f(x(t))+d(x(t)):$$ $$\lim_{\hat{g}(t) \to c^{-}} \frac{d}{dt}\hat{g}(t)>0 \quad \text{and} \ \ \ \ \lim_{\hat{g}(t) \to c^{+}} \frac{d}{dt}\hat{g}(t)<0$$ under the condtion $$|d(x(t))|<f(x(t))$$ Also, I would like to receive a good reference to such type of limit theory.
Thanks!