All Mutation Rates $c\mathrm{/}n$ for the $(1+1)$ Evolutionary Algorithm

For every real number $c\ge 1$ and for all $\epsilon >0$, there is a fitness function $f:\{0,1{\}}^n\to \mathbb{R}$ for which the optimal mutation rate for the $(1+1)$ evolutionary algorithm on $f$, denoted $p_n$, satisfies $p_n\approx c\mathrm{/}n$ in that $\mathrm{\mid }n{p}_n-c\mathrm{\mid }<\epsilon$. In other words, the set of all $c\ge 1$ for which the mutation rate $c\mathrm{/}n$ is optimal for the $(1+1)$ EA is dense in the interval $[1,\mathrm{\infty })$. To show this, a fitness function is introduced which is called HillPathJump.