I have two questions.

i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution $f:\mathbb{R}\to\mathbb{R}$ for all $x\neq y$?

ii) Let $\varphi(x)=x$.

Does the functional equation $$ |f(x)-f(y)|=\frac{1}{|x-y|} $$ have a solution for all $x\neq y$?

Great thanks in advance!