3.5 Uniform Continuity 83 4 DIFFERENTIATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1 Definition and Basic Properties of the Derivative 89 4.2 The Mean Value Theorem 95 4.3 Some Applications of the Mean Value Theorem 102 4.4 L’Hôpital’s Rule 104 4.5 Taylor’s Theorem 111 5 ADDITIONAL TOPICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Topology of the Real Line 115 5.2 Continuity and Compactness 122 5.3 Limit Superior and Limit Inferior of Functions 125 5.4 Lower Semicontinuity and Upper Semicontinuity 131 5.5 Convex Functions and Derivatives 136 5.6 Nondifferentiable Convex Functions and Subdifferentials 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Solutions and Hints for Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 153
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