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 \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_{\text{eps}}$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_{\text{eps}}$ be the corresponding invariant density. We prove that $\text{eps}\mapsto h_{\text{eps}}$ is differentiable in $L^1$ and provide a formula for its derivative.