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Almost all the theorems in this book are well-known old results of a carefully studied subject. The well-known ones are more important than the few novel aspects of the book.

However, some details like the converse of Taylor's theorem (both continuous and discrete) are not so easy to nd in traditional calculus sources. The microscope theorem for dierential equations does not appear in the literature as far as we know, though it is similar to research work of Francine and Marc Diener from the 1980s. We conclude the book with convergence results for Fourier series. While there is nothing novel in our approach, these results have been lost from contemporary calculus and deserve to be part of it. Our development follows Courant's calculus of the 1930s giving wonderful results of Dirichlet's era in the 1830s that clearly settle some of the convergence mysteries of Euler from the 1730s. This theory and our development throughout is usually easy to apply. Clean theory should be the servant of intuition building on it and making it stronger and clearer.

There is more that is novel about this book. It is free and it is not a book since it is not printed. Thanks to small marginal cost, our publisher agreed to include this electronic text on CD at no extra cost. We also plan to distribute it over the world wide web. We hope our fresh look at the foundations of calculus will stimulate your interest. Decide for yourself what's the best way to understand this wonderful subject. Give your own proofs.

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