For f(t)(x) = t-x(2) the quadratic family, we define the fractional susceptibility function Psi(Omega)(phi,t0) (eta, z) of f(t), associated to a C-1 observable phi at a stochastic parameter t(0). We also define an approximate, "frozen," fractional susceptibility function Psi(fr)(phi,t0) (eta, z) such that lim(eta -> 1) Psi(fr)(f,t0) (eta, z) is the susceptibility function Psi(f,t0) (z) studied by Ruelle. If t(0) is Misiurewicz-Thurston, we show that Psi(fr)(phi,t0) (1/2, z) has a pole at z = 1 for generic phi if J(1/2)(t(0)) not equal 0, where J(eta)(t) = Sigma(infinity)(k=0) sgn(Df(t)(k) (c(1)))vertical bar Df(t)(k) (c(1))vertical bar(-eta), with c(1) = t the critical value of f(t). We introduce "Whitney" fractional integrals I-eta,I-Omega and derivatives M-eta,M-Omega on suitable sets Omega. We formulate conjectures on Psi(phi,t0)Omega (eta, z) and J(eta)(t), supported by our results on M-eta,M-Omega and Psi(fr)(phi,t0) (1/2, z), for the former, and numerical experiments, for the latter. In particular, we expect that Psi(Omega)(phi,t0) (1/2, z) is singular at z = 1 for Collet-Eckmann t(0) and generic phi. We view this work as a step towards the resolution of the paradox that Psi(phi,t0) (z) is holomorphic at z = 1 for Misiurewicz-Thurston f(t0) (Jiang and Ruelle in Nonlinearity 18:2447-2453, 2005, Ruelle in Commun Math Phys 258:445-453, 2005), despite lack of linear response