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Chapter 9: Uniform Convergence

In Chapter 5 we touched on the idea of convergent sequences of functions. In this chapter we develop the idea a lot further. It turns out that many (but not all!) important properties that a sequence of functions might are also inherited by its limit. This includes integrability and continuity, but not differentiablility. The whole idea of limits of functions leads to consideration of approximation of functions by other functions (a topic that's important in numerical analysis) and we prove that all continuous functions can be approximated by polynomials.

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