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Basic Analysis I: Continuity and Smoothness

One good way to generate errors and embarrass yourself is to use a formula or identity without properly verifying its validity under the circumstances at hand. This seems particularly easy to do in Fourier analysis, and it is not at all unusual to see differential identities and integral formulas from the theory of Fourier analysis being used with functions that are neither differentiable or integrable. (It’s especially disturbing when such abuses occur in textbooks.) The results range from questionable to disastrously wrong.

We, of course, will try to avoid such mistakes. So we must be able to identify when the various results derived in this text are valid and when they are not. To simplify this ...

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