  For the quadratic family, we define the two-variable ($\eta$ and $z$)
fractional susceptibility function associated to a C^1 observable at a
stochastic map. We also define an approximate, "frozen" fractional
susceptibility function. If the parameter is Misiurewicz-Thurston, we show that
the frozen susceptibility function has a pole at $z=1$ for generic observables
if a "one-half" transversality condition holds. We introduce "Whitney"
fractional integrals and derivatives on suitable sets $\Omega$. We formulate
conjectures supported by our results on the frozen susceptibility function and
numerical experiments. In particular, we expect that the fractional
susceptibility function for $\eta=1/2$ is singular at $z=1$ for Collet-Eckmann
maps and generic observables. We view this work as a step towards the
resolution of the paradox that the classical susceptibility function is
holomorphic at $z=1$ for Misiurewicz-Thurston maps, despite lack of linear
response.
