I’m looking at the definition and can’t wrap my head around
In: Mathematics
Continuity means that given epsilon, for any point a on the function’s domain, I can find a delta such that|x-a|<delta –> |f(x)-f(a)| < epsilon. In other words, delta may depend on the specific value of a that I’m looking at. So we can think of delta as dependent on a. I like to write it delta(a) as a reminder that it’s really a function.
Now, uniform continuity just means that delta(a) is a constant. Ie, it does not depend on the choice of a. In other words, given epsilon, I can find a delta such that |x-a|< delta –> |f(x)-f(a)| *for any* a.
Notice that in continuity, I choose a first, then find a delta that works for that specific a, whereas in uniform continuity I get to pick delta first, since it works for any and all values of a.
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