How to prove uniform continuity from bounded derivatives?
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Basically how my book distinguishes this from point-wise continuity is that there exists a single δδthat works for every point in the domain, so once we find that δδ, we know it works everywhere. On the other hand, point-wise continuity says that given a c∈Ac∈A, there exists a δδ such that the function is continuous at cc, but all these δδs may be different, perhaps depending on cc, and we might not be able to find just one δδ that works for all ccs everywhere.
I interpret this definition a completely different way, and I want to see if my conjecture is correct. I think that functions which have bounded derivatives are uniformly continuous. That is, if the "steepness" and "shallowness" of a function is limited to a certain minimum and maximum, then the there's a sufficiently small enough δδ that we can use, particularly at the steepest part of the function (say at x0x0), such that the output stays within the εε-neighborhood of f(x0)f(x0).
I interpret this definition a completely different way, and I want to see if my conjecture is correct. I think that functions which have bounded derivatives are uniformly continuous. That is, if the "steepness" and "shallowness" of a function is limited to a certain minimum and maximum, then the there's a sufficiently small enough δδ that we can use, particularly at the steepest part of the function (say at x0x0), such that the output stays within the εε-neighborhood of f(x0)f(x0).
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