Does this type of phase transition exist? The short version of this question is:


*

*Is there, or could there be, a system with a phase transition where adding a small amount of heat causes a discontinuous jump in its temperature?


Below are my reasons for thinking there might be.
At a first-order phase transition there is a discontinuity in the first derivative of $\log Z(\beta)$, where $\log Z$ is essentially the free energy and $\beta=1/k_B T$ is the inverse temperature. As a consequence, its Legendre transform $S(E)$ has a segment of zero second derivative. (Here $S$ is the entropy and $E$ the expected value of the energy. The two functions are related by $S(E) = \log Z(\beta) + \beta E$.) This means that the function $E(\beta)$ has a discontinuity. These basic properties of first-order phase transitions are illustrated below:

The slope of the third plot, $d E/d\beta$, is related to the heat capacity, which becomes infinite when $\beta$ is at the critical value.
However, it seems to me that the opposite phenomenon could also happen, where the discontinuity is in the first derivative of $S(E)$, and consequently $\log Z(\beta)$ has a straight line segment, and $E(\beta)$ has a section of zero rather than infinite slope, like this:

Rather than having an infinite heat capacity for a critical value of $\beta$, such a material would have a zero heat capacity for a critical value of its energy density, meaning that at the critical point, adding a small amount of energy would cause a discontinuous change in temperature.
It seems that it wouldn't be too hard to construct a toy model that exhibits this "dual" type of phase transition. All you really need is a very high density of states at the critical energy value. (However, I have not explicitly constructed such a model yet.)
In a similar way, one could construct the dual of a continuous phase transition. Here the second derivative of $S(E)$ would diverge at the critical point, and the heat capacity would smoothly approach zero around the transition.
I have never seen anyone refer to these types of transition, but I don't know whether this is because (a) they don't happen, (b) they're not considered very interesting, or (c) I just don't know the correct term for this phenomenon. Therefore my questions are


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*Does this type of transition occur in physical systems? If so, does this transition type have a name, and is there a well-studied example?

*If not, is there a fundamental reason why it can't happen? What assumptions are needed to prove that it can't?
 A: I think I can answer the question. Such processes are possible. The only example I know is latent heat release of a supercooled phase. (You may search for "supercooled water" on youtube).
The interesting feature, i.e. the sudden jump of the temperature without a change of energy is well documented e.g. in Braga and Milon 2012, see Fig. 7 top rows. During one second, the temperature of supercooled water changes from -10°C to 0°C. 
Such phase changes also occur solid to solid, referred to as Martensitic transformations in crystals. Again, one has to supercool the high temperature phase below the heterogeneous nucleation temperature. Upon perturbation, the higher symmetric crystal lattice of the Austenitic (high temperature) phase decomposes into some microstructure of lower symmetric crystal lattices of the Martensitic (low temperature) phase. However, the temperature increase in these cases is in general of minor importance, the mechanical effects are much more interesting. 
So, from your options a), b) and c), I think c is the answer.
PS: Engineers that design phase-change based latent heat storages still do not need to consider the dynamics of such a process, all they need to know is the enthalpy of fusion of the material. The energy that can be stored in the supercooled liquid tank is just the heat of fusion in J/g times the mass. Nucleation and the nearly instantaneous phase change transform the stored heat of fusion into an actual temperature increase, which is then used to drain energy by ordinary heat flow from the tank.
A: Interesting Question. I just recently observed this kind of Phase transition; 

in the Thermosphere of the Earth, at about 117 km high. At that point the Matter turns from Neutral to Electrically charged ions. So it's also some new not previously known plasma-state of matter. 
When I look your Enropy (S) -Energy (E) and, Free Energy (Log Z) - inverse Temperature ($\beta$) diagrams, This must fit mainly to your first example. 
It might just need to be written $\beta=1/k_BTP$ ,which means that also Pressure is considered.
