  We consider the susceptibility function Psi(z) of a piecewise expanding
unimodal interval map f with unique acim mu, a perturbation X, and an
observable phi. Combining previous results (deduced from spectral properties of
Ruelle transfer operators) with recent work of Breuer-Simon (based on
techniques from the spectral theory of Jacobi matrices and a classical paper of
Agmon), we show that density of the postcritical orbit (a generic condition)
implies that Psi(z) has a strong natural boundary on the unit circle. The
Breuer-Simon method provides uncountably many candidates for the outer
functions of Psi(z), associated to precritical orbits. If the perturbation X is
horizontal, a generic condition (Birkhoff typicality of the postcritical orbit)
implies that the nontangential limit of the Psi(z) as z tends to 1 exists and
coincides with the derivative of the acim with respect to the map (linear
response formula). Applying the Wiener-Wintner theorem, we study the
singularity type of nontangential limits as z tends to e^{i\omega}. An
additional LIL typicality assumption on the postcritical orbit gives stronger
results.
