Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope-an influential notion of steepest descent in a nonsmooth setting. The existence theory is further amplified for semialgebraic functions, prototypical nonpathological functions in nonsmooth optimization: such functions always admit nontrivial descent curves emanating from any (even critical) nonminimizing point. We moreover show that curves of near-maximal slope of semialgebraic functions have a more classical description as solutions of subgradient dynamical systems.