Limits and Continuity
Having considered the fundamental properties of the real numbers in Chapters 1 and 2, we now commence our study of real analysis proper. The heart of real analysis, and one of the key features that distinguishes real analysis from algebraic and combinatorial branches of mathematics, is the concept of a limit. There are various types of limits, for example limits of functions (which we will discuss in the present section), and limits of sequences (to be discussed in Section 8.2). However, all of these types of limits have similar features, and gaining familiarity with one type of limit will make learning about the other types much easier. The reader has already encountered limits of functions in an intuitive fashion in calculus courses. However, limits take on a much more important role in real analysis than in calculus because in the former we are concerned with rigorous proofs rather than applications, and limits are at the heart of the rigorous treatment of calculus.
KeywordsRational Number Open Interval Real Analysis Discontinuous Function Uniform Continuity
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