Linear response due to singularities

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a cusp at the turning point, we recover the linear response. More precisely, let T_ɛ be a family of such cusp maps generated by changing the value of the turning point of T₀ by a deterministic perturbation and let h_ɛ be the corresponding invariant density. We prove that𝜀↦ℎ𝜀 is differentiable in L¹ and provide a formula for its derivative.