If by continuity you mean, "a feeling that I am who I was before the operation, (perhaps with some changes)", then it seems that each hemisphere would separately maintain continuity, in the same way patients after massive strokes and other sudden brain injuries don't usually feel "they are a different person".
Research by Turk et al. (2003) suggests it's just the left hemisphere, because only the left hemisphere has the 'interpreter' module.
However, other research by Turk et al. (2002) find that both hemispheres of a split brain patient recognize his own face, similar to Sperry et al. (1979). I think these face recognition tests tell us less about the 'feeling of self' than tests of access to autobiographical memories before the operation, and their emotional content, but I couldn't find any such studies.
A good review of the field instead was done by Uddin in 2011.
Sperry, R. W., Zaidel, E., & Zaidel, D. (1979). Self recognition and social awareness in the deconnected minor hemisphere. Neuropsychologia, 17(2), 153–166. doi:10.1016/0028-3932(79)90006-X
Turk, D. J., Heatherton, T. F., Kelley, W. M., Funnell, M. G., Gazzaniga, M. S., & Macrae, C. N. (2002). Mike or me? Self-recognition in a split-brain patient. Nature Neuroscience, 5(9), 841–842. doi:10.1038/nn907
Turk, D. J., Heatherton, T. F., Macrae, C. N., Kelley, W. M., & Gazzaniga, M. S. (2003). Out of Contact, Out of Mind. Annals of the New York Academy of Sciences, 1001(1), 65–78. doi:10.1196/annals.1279.005
Uddin, L. Q. (2011). Brain connectivity and the self: The case of cerebral disconnection. Consciousness and Cognition, 20(1), 94–98. doi:10.1016/j.concog.2010.09.009