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Introduction to Real Analysis
Dr. Weihu Hong Clayton State University 11/4/2008
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Continuous Functions Definition Let E be a subset of R and f a real-valued function with domain E, that is, f : ER. 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.
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Continuous Functions Remark. The function f is continuous at a point p in E if and only if
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Make Continuous Functions From Old Ones
Theorem Let E be a subset of R and f and g are real-valued functions with domain E, that is, f : ER, g : ER. Assume both functions f and g are continuous at a point p in E, then
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Make Continuous Functions From Old Ones
Theorem
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Topological Characterization of Continuity
Theorem
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Continuity and Compactness
Theorem Corollary
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Intermediate Value Theorem (IVT)
Corollary
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Intermediate Value Theorem (IVT)
Corollary Corollary (A fixed point).
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