The Balmer break comes from a combination of two main things: the ability of photons with high enough energies (wavelengths shorter than 364.5 nm) to ionize hydrogen atoms that are in the $n = 2$ energy level, and the inability of any photons with lower energies (wavelengths longer than 364.5 nm) to ionize the atoms. This leads to an abrupt change in absorption of outgoing photons, from high absorption at wavelengths shorter than 364.5 nm, to no absorption at longer wavelengths.
Another important factor: Any photon with a wavelength shorter than 364.5 nm can ionize an $n = 2$ hydrogen atom -- but the probability (due to the cross-section of the photon-atom interaction) goes down as the wavelength gets shorter. That's why you get the sawtooth pattern: very-short-wavelength photons can ionize the atoms, but are unlikely to. As the wavelengths get longer, the probability of ionizing a photon increases, up to the point of 364.5 nm. Once the wavelength is longer than that, ionization is simply impossible, so the absorption goes to zero.
In practice, the total absorption doesn't go to zero, because there are other sources of absorption, including hydrogen atoms in the higher states, like the $n = 3$ state (which has its own "Paschen jump"), and things like H$^{-}$ hydrogen ions. You also have to think about things like how many atoms are in different energy levels. In a very hot atmosphere, most of the hydrogen atoms will be in higher energy levels (or even ionized), so you won't get much of a Balmer break.
do bound-bound absorptions not occur at all for stars with these discontinuities?
They absolutely do occur -- they produce absorption lines, such as those of the Balmer series (which are not shown in the simplified, cartoon spectra in your figure). If you have enough hydrogen atoms in the $n = 2$ state to produce a noticeable Balmer break, then you'll automatically have Balmer absorption lines (H$\alpha$, H$\beta$, etc.) due to longer-wavelength photons being absorbed by the same atoms.
These lines do get closer together as you approach 364.5 nm from the long-wavelength side, which has the effect of making the peak of the sawtooth a bit rounded, but they don't produce the jump -- they make it a little bit more gradual. (Your figures do not show the effects of the absorption lines.)