### On cusps and flat tops

Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to ${C}^{1+\u03f5}$. The critical points are not required to verify a non-flatness condition, so the results are applicable to ${C}^{1+\u03f5}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.