  We consider a family of Pomeau-Manneville type interval maps $T_\alpha$,
parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous
invariant probability measures $\nu_\alpha$, and rate of correlations decay
$n^{1-1/\alpha}$. We show that despite the absence of a spectral gap for all
$\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geq 1/2$,
the map $\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously
differentiable for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large.
