KöMaL 2024 A895

Let's call a function $f\colon\mathbb{R}\to\mathbb{R}$ weakly periodic if it is continuous and satisfies ${f(x+1)}=f(f(x))+1$ for all $x\in \mathbb{R}$. a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb{R}$? b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb{R}$?