Linear response for systems with a cusp

In this note we consider a tent-like family with a cusp at the singular point and show that the linear response holds for certain perturbations of this family. This contrasts the tent-like maps with finite derivatives at the singularity. Our results extend the results of Bahsoun and Galatolo to the larger class of singularities and we obtain the linear response formula in $L^p$ for $p>1$.