Query - as I understand it general relativity is based on spacetime being continuous while in quantum physics it has discontinuities. Has there been any theories developed to somehow come up with a theory to cover both the continuity and discontinuities?
Assuming we are just talking about the space-time both theories use as the OP stated- this has nothing to do with the energy states of particles.
If anything General relativity is the one with 'discontinuities' because it allows things like singularities and closed time-like curves. Quantum mechanics on the other hand has a sort of 'backdrop' spacetime - a simple Euclidean 'stage' that the particles act out their interactions, so I'd say that QM space-time was actually always handled as continuous.
However it may be that you are thinking about the idea of the Planck-length spacetime, that makes a smallest 'bit' of QM spacetime because of fundamental limits at trying to measure spacetime at those levels (i.e. we end up expending so much energy that we create black holes.) Whereas GR does not have such a problem as it doesn't model the microscopic at all.
Now there are some theories that assume that is the reality - that the universe is effectively made up of these discrete bits and they are real, possibly even observable if we are in fact in tha hologram universe (but that's another story...) So then the universe is some sort of giant QM computer built from these bits. But other QM theories still assume that space-time is continuous, just that on scales below the Planck length are completely inaccessible for measurement.
However remember both theories are incomplete. General relativity breaks down on the small scale and cannot explain QM observatoins, plus it forms these singularities which my guess is, will disappear in a unified theory. Whilst Quantum mechanics can't handle the gravity force nor the large scale structure questions.
A new theory of everything will hopefully shed light on what is a better model of reality and whether space-time is continuous or discontiuous, as it should provide a new model.
EDIT: i.e. Some approaches such as
Causal Dynamical Trigulations have a fascinating properties that on the extremely small scale, spacetime for quantum particles drops from fourth dimensions to two. My basic understanding is that the two 'extra' dimensions do not dissappear, btw, so it's not really a discontinuity in a 'real' sense, but more of a observational/measurement thing. Interestingly other approaches come to the same conclusions with respect to this dimensionality.