My impression always was that this is because of ingrained mathematical culture of seeking the most precise requirements for a particular theorem to hold. That way, when a non-smooth situation eventually does emerge, the theorems are already in place to deal with it. It's one of the differences in culture between pure mathematics, applied mathematics and physics.
I always wondered about looking at the problem the other way around: if you assume that stronger and stronger continuity holds, then what extra properties hold? What about $C^{infty}$ cases and analytic, is there a gap in between?
The same phenomenon appears in a lot of the optimization literature, where weaker and weaker continuity requirements are made on the functions being optimized. I find this weird -- why not instead create faster and faster optimizers that leverage the smoothness properties present in most applications?
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