The reactions $gamma pto eta p$ and $gamma pto eta p$ have been measured from their thresholds up to the center-of-mass energy $W=1.96$GeV with the tagged-photon facilities at the Mainz Microtron, MAMI. Differential cross sections were obtained with unprecedented accuracy, providing fine energy binning and full production-angle coverage. A strong cusp is observed in the total cross section and excitation functions for $eta$ photoproduction at the energies in vicinity of the $eta$ threshold, $W=1896$MeV ($E_gamma=1447$MeV). This behavior is explained in a revised $eta$MAID isobar model by a significant branching of the $N(1895)1/2^-$ nucleon resonance to both, $eta p$ and $eta p$, confirming the existence and constraining the properties of this poorly known state.