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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.
## Chapter Contents