When thinking about agents, the first motivation might not quite work out. Small changes in observation might introduce discontinuous changes in policy—e.g. in the Matching Pennies game. Suppose there are agents (functions) in X that output a fixed P(Heads), no matter their input. If you can continuously vary P(heads) by moving in X, then Matching Pennies play will be discontinuous at P(Heads)=0.5. So right away you’ve committed to some unusual behavior for the agents in X by asking for continuity—they can’t play perfect Matching Pennies at the very least.
When thinking about agents, the first motivation might not quite work out. Small changes in observation might introduce discontinuous changes in policy—e.g. in the Matching Pennies game. Suppose there are agents (functions) in X that output a fixed P(Heads), no matter their input. If you can continuously vary P(heads) by moving in X, then Matching Pennies play will be discontinuous at P(Heads)=0.5. So right away you’ve committed to some unusual behavior for the agents in X by asking for continuity—they can’t play perfect Matching Pennies at the very least.