KöMaL 2019 A756

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ (functions with domain $\mathbb{R}$ and values from $\mathbb{R}$) which satisfy the following two conditions: $(i)$ $f(x+1)=f(x)+1$; $(ii)$ $f(x^2)=\big(f(x)\big)^2$.