  We explore the quantum phase transitions between two ordered states in the
infinite dimensional Hubbard-Holstein model at half filling. Our study is based
on the dynamical mean field theory (DMFT) combined with the numerical
renormalization group (NRG), which allows us to handle both strong
electron-electron and strong electron-phonon interactions. The transition line
is characterized by an effective electron-electron interaction. Depending on
this effective interaction and the phonon frequency $\omega_0$ one finds either
a continuous transition or discontinuous transition. Here, the analysis focuses
on the behavior of the system when the electron-electron repulsion $U$ and the
phonon-mediated attraction $\lambda$ are equal. We first discuss the adiabatic
and antiadiabatic limiting cases. For finite $\omega_0$ we study the
differences between the antiferromagnetic (AFM) and charge order, and find that
when present the AFM state has a lower energy on the line.
