We theoretically predict the stability of liquid in a model brush made of flexible fibers for cases in which liquid is supplied from an ink reservoir. The volume of the liquid in the brush increases with increasing applied pressure by the reservoir, and the liquid shows instability at a critical pressure. When the fibers are shorter than a critical length, the end of the brush opens continuously with increasing applied pressure. The volume of the liquid that hangs from the open end of the brush increases with increasing applied pressure, and the liquid drops from the brush at the critical pressure, where the weight of the liquid becomes larger than the surface tension. In contrast, when the fibers are longer than the critical length, the end of the brush opens discontinuously to the maximal extent at the critical pressure. The discontinuous unbuckling is driven by the instability arising from the fact that the bending stiffness of the water surface, which bends together with the flexible fibers, decreases as the end of the brush opens, and it is thus a unique feature of brushes of flexible fibers.