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Unformatted text preview: MATH 2400 LECTURE NOTES: CONTINUITY AND LIMITS PETE L. CLARK Contents 1. Introduction to Calculus 1 1.1. Some derivatives without formal limits 1 2. Some derivatives without a careful definition of limits 3 3. Limits in Terms of Continuity 5 4. Continuity Done Right 7 4.1. The formal defininition of continuity 7 4.2. Basic properties of continuous functions 8 5. Limits Done Right 11 5.1. The Formal Definition of a Limit 11 5.2. Basic Properties of Limits 12 5.3. The Squeeze Theorem and the Switching Theorem 13 5.4. Variations on the Limit Concept 16 References 18 1. Introduction to Calculus 1.1. Some derivatives without formal limits. We have seen that in order to define the derivative f of a function f : R R we need to understand the notion of a limit of a function at a point. It turns out that giving a mathematically rigorous and workable definition of a limit is hard really hard. Let us begin with a quick historical survey of this problem. It is generally agreed that calculus was invented (discovered?) independently by Isaac Newton and Gottfried Wilhelm von Leibniz, roughly in the 1670s. Leibniz was the first to publish on calculus, in 1685. However Newton probably could have published his work on calculus before Leibniz, but held it back for various reasons. 1 As usual, saying calculus was discovered by Newton and Leibniz is an over- simplification. Computations of areas and volumes which we can now recognize as using calculus concepts go back to ancient Egypt, if not earlier. The Greek math- ematicians Eudoxus (408-355 BCE) and Archimedes (287-212 BCE) developed the method of exhaustion , a limiting process which anticipates integral calculus. Also Chinese and Indian mathematicians made significant achievements. Even in 1 I highly recommend James Gleicks biography of Newton. If I wanted to distill hundreds of pages of information about his personality into one word, the word I would choose is... weirdo . 1 2 PETE L. CLARK modern Europe Newton and Leibniz were not functioning in an intellectual vac- uum. They were responding to and continuing earlier work by Pierre de Fermat (on tangent lines, for instance) and John Wallis, Isaac Barrow and James Gregory. But this should not be surprising: all scientific and intellectual work builds on work of others. The point is that the accomplishments of Newton and Leibniz were sig- nificant enough so that after their efforts calculus existed as a systematic body of work, whereas before them it did not. How did Newton and Leibniz construe the fundamental concept, namely that of a limit? In fact both of their efforts were far from satisfactory, indeed far from making good sense. Newtons limiting concept was based on a notion of fluxions , which is so obscure that we need not say anything about it here. Leibniz, by his nature a philosopher and writer as well as a mathematician, at least addressed the diculties less obscurely and came up with the notion of an infinitesimal quan-...
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