Rigorous treatment of topics introduced in calculus such as continuity, uniform convergence, power series, differentiation, and integration. Development of mathematical proof in these contexts. Sequence with MATH 316.
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Expanded Course Description
This is the beginning of a huge subject called real analysis, also called function theory. In high school mathematics one learns a few basic functions (polynomials, exponentials, logarithms, trigonometric functions...) but this is only the tip of an iceberg. There are lots of other important functions even in applications of mathematics to the sciences that cannot be written down except by using a limiting process. Analysis is the study of such functions, which can turn out to be far more complicated than anyone first imagines.
MATH 317 focuses on techniques for making new functions out of old ones via limiting processes. When a new function is made in this way one needs to answer basic questions like: Is it continuous? Does it have a derivative? What happens when we integrate it? We will learn the main results that allow us to answer such questions. Along the way we will see rigorous treatments of differentiation and integration, as well as many examples of really crazy functions one can create out of very simple processes. The culmination of the course is a completely rigorous treatment of exponential and logarithm functions.
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