When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the Hamiltonian does not close the gap (or break symmetries in the case of symmetry-protected topological phases).
This is clear when we limit the ways a hamiltonian can change only to a few scalar or vector parameters (hopping parameters in tight-binding models, magnetic and electric field magnitudes or directions). Then one could say that the the transformation is continuous if for instance in a given basis the matrix elements of the hamiltonian are continuous functions of the parameters (which has the usual definition for complex/real valued functions).
In general one says that a function is continuous on some topological space if the pre-images of open sets are open sets. This requires a topology on the space. In this case the space of linear operators, possibly limited to hermitian operators or an even smaller space. This space has a lot of structure and there are for instance various ways to define limit of operators (I only know the limits in the weak and strong sense, but I know there are other definitions, never seen used in physics).
Does then "continuous" mean that we choose some topology on our space of hamiltonians (whatever this may be - perhaps not even hermitian is needed in general) and the definition is as usual? If yes, is there a "usual" choice for the topology, derived from some other property of the space? If not, what does continuous mean in this case?