Let us say that a bounded smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$ has *vanishing variation at infinity* (or satisfies "property $A$" for short) if, for any $r\neq 0$, we have

$$\lim_{x\rightarrow\infty}\frac{|f(x+r)-f(x)|}{|r|} = 0.$$

In particular this means that

$$\lim_{r\rightarrow 0}\left(\lim_{x\rightarrow\infty}\frac{|f(x+r)-f(x)|}{|r|}\right) = 0.$$

An example of such a function is any smooth function whose *gradient vanishes at infinity* (or satisfies "property $B$" for short): that is,

$$\lim_{x\rightarrow\infty}\left(\lim_{r\rightarrow 0}\frac{|f(x+r)-f(x)|}{|r|}\right) = 0.$$

On the other hand, there are functions satisfying property $B$ that do not satisfy property $A$. One type of example is any smooth function that has a limit as $x\rightarrow\infty$ but whose derivative does not go to zero; however note that such functions can always be approximated in the uniform norm by a sequence of functions satisfying property $A$.

**Question**: can *any* function satisfying $A$ be approximated in the uniform norm by a sequence of functions satisfying $B$? More precisely: given any $f$ satisfying $A$, can we find a sequence of functions $(f_i)$ satisfying $B$ such that for any $\epsilon > 0$, $||f-f_i||_\infty<\epsilon$ for all $i$ sufficiently large?