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Introduction to Real Analysis

Introduction to Real Analysis. K.VIMALA DEPARTMENT OF MATHEMATICS. 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

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Introduction to Real Analysis

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  1. Introduction to Real Analysis K.VIMALA DEPARTMENT OF MATHEMATICS

  2. 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.

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

  4. 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

  5. Make Continuous Functions From Old Ones • Theorem 4.2.4.

  6. Topological Characterization of Continuity • Theorem 4.2.6.

  7. Continuity and Compactness • Theorem 4.2.8. • Corollary 4.2.9.

  8. Intermediate Value Theorem (IVT) • Theorem 4.2.11 (IVT). • Corollary 4.2.12.

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

  10. THANK YOU

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