  Jump functions are the {most-studied} non-unimodal benchmark in the theory of
randomized search heuristics, in particular, evolutionary algorithms (EAs).
They have significantly improved our understanding of how EAs escape from local
optima. However, their particular structure -- to leave the local optimum one
can only jump directly to the global optimum -- raises the question of how
representative such results are.
  For this reason, we propose an extended class $\textsc{Jump}_{k,\delta}$ of
jump functions that contain a valley of low fitness of width $\delta$ starting
at distance $k$ from the global optimum. We prove that several previous results
extend to this more general class: for all {$k \le \frac{n^{1/3}}{\ln{n}}$} and
$\delta < k$, the optimal mutation rate for the $(1+1)$~EA is
$\frac{\delta}{n}$, and the fast $(1+1)$~EA runs faster than the classical
$(1+1)$~EA by a factor super-exponential in $\delta$. However, we also observe
that some known results do not generalize: the randomized local search
algorithm with stagnation detection, which is faster than the fast $(1+1)$~EA
by a factor polynomial in $k$ on $\textsc{Jump}_k$, is slower by a factor
polynomial in $n$ on some $\textsc{Jump}_{k,\delta}$ instances.
  Computationally, the new class allows experiments with wider fitness valleys,
especially when they lie further away from the global optimum.
