If you have surreal-valued utilities, you can just round infinitesimals to 0 to get real-valued utilities, and then continuity can be satisfied with real-valued probabilities again. The resulting real-valued utility function is correct about your preferences whenever it assigns higher utility to one option than the other, and is deficient only in the case where it assigns the same utility to two different options that you value differently. But it is very unlikely for two arbitrary reals to be exactly the same, and even when this does happen, the difference is infinitesimally unimportant compared to other preferences, so this isn’t a big loss.
If you have surreal-valued utilities, you can just round infinitesimals to 0 to get real-valued utilities, and then continuity can be satisfied with real-valued probabilities again. The resulting real-valued utility function is correct about your preferences whenever it assigns higher utility to one option than the other, and is deficient only in the case where it assigns the same utility to two different options that you value differently. But it is very unlikely for two arbitrary reals to be exactly the same, and even when this does happen, the difference is infinitesimally unimportant compared to other preferences, so this isn’t a big loss.