Introduction to Real Analysis

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Presentation transcript:

Introduction to Real Analysis Dr. Weihu Hong Clayton State University 11/4/2008

Continuous Functions Definition 4.2.1. Let E be a subset of R and f a real-valued function with domain E, that is, f : ER. The function f is continuous at a point p in E, if The function f is continuous on E if and only if f is continuous at every point of E.

Continuous Functions Remark. The function f is continuous at a point p in E if and only if

Make Continuous Functions From Old Ones Theorem 4.2.3. Let E be a subset of R and f and g are real-valued functions with domain E, that is, f : ER, g : ER. Assume both functions f and g are continuous at a point p in E, then

Make Continuous Functions From Old Ones Theorem 4.2.4.

Topological Characterization of Continuity Theorem 4.2.6.

Continuity and Compactness Theorem 4.2.8. Corollary 4.2.9.

Intermediate Value Theorem (IVT) Corollary 4.2.12.

Intermediate Value Theorem (IVT) Corollary 4.2.13 Corollary 4.2.14 (A fixed point).