  We present lattice results for the vector and scalar form factors of the
semileptonic decays D -> \pi \ell \nu_ell and D -> K \ell \nu_\ell in the
physical range of values of squared four momentum transfer q^2, obtained with
N_f=2 maximally twisted Wilson fermions simulated at three different lattice
spacings (a ~ 0.102 fm, 0.086 fm, 0.068 fm) with pion masses as light as 270
MeV and m_\pi L \gtrsim 4. The form factors are extracted using a double ratios
strategy, which allows a good statistical accuracy and is independent of the
vector current renormalization constant. The chiral/continuum extrapolation is
performed through a simultaneous fit in the three variables (m_\pi, q^2, a)
using HMChPT formulae with additional O(a^2) terms that parametrically account
for the lattice spacing dependence. Our results are in very good agreement with
the experimental data in the full q^2 range for both D -> \pi \ell \nu_\ell and
D -> K \ell \nu_\ell. At zero momentum transfer we obtain f^{D->pi}(0) =
0.65(6)_{stat}(6)_{syst} and f^{D->K}(0) = 0.76(5)_{stat}(5)_{syst}, where the
systematic error does not include the effects of quenching the strange and the
charm quarks. Our findings are in good agreement with recent lattice
calculations at N_f = 2+1.
