Introduction to Mathematical Analysis I - Second Edition

3.4 PROPERTIES OF CONTINUOUS FUNCTIONS 77 3.5 UNIFORM CONTINUITY 83 3.6 LIMIT SUPERIOR AND LIMIT INFERIOR OF FUNCTIONS 89 3.7 LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY 94 4 DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 DEFINITION AND BASIC PROPERTIES OF THE DERIVATIVE 101 4.2 THE MEAN VALUE THEOREM 107 4.3 SOME APPLICATIONS OF THE MEAN VALUE THEOREM 114 4.4 L’HOSPITAL’S RULE 117 4.5 TAYLOR’S THEOREM 123 4.6 CONVEX FUNCTIONS AND DERIVATIVES 126 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS 132 5 Solutions and Hints for Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 143