There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it.
Playing with the definition, I want to say uniform continuity implies there’s a maximum “average rate of change”. Not literally a derivative, but the rate of chance between two points is bounded in the domain. I’m aware that this is essentially Lipschitz continuity, and that Lipschitz implies uniform. This implies there’s more to uniform continuity than just having a bounded average rate of chance.
And also, how is it that f(x)=x is uniform yet f(x)f(x)=g(x)=x^2 is not? I understand why it isn’t, I can prove it. But I just don’t understand the motivation and important of uniform continuity.
from Hot Weekly Questions - Mathematics Stack Exchange
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