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Chapter 4: Differentiable and Continuous Functions on the Real Line

Differentiation involves taking the limit of a function of a continuously varying quantity, so it involves a different kind of limit from sequences. We start by studying limits of this kind, and introduce limits of functions and continuity. Nevertheless, Proposition 4.2 shows how to relate a lot of this to the results of the last chapter on seuences. In this chapter we'll also prove the basic rules of calculus for computing derivatives: the Chain rule, the product rule, and the quotient rule.

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