紧集上的连续映射 为了将一元连续函数在闭区间上的重要性质推广到多元连续函 数，为此先定义多元函数在点集的边界点连续的概念。 定义 11.3.1 设点集 K ⊂ n R ，f : K→ m R 为映射（向量值函数）， x K 0 ∈ 。如果对于任意给定的ε > 0，存在δ > 0，使得当 0 xx K ∈O( ,) δ ∩ 时
1 Preliminary 11 Introduction 12 Cardinality 13 Topology of the Euclidean space 14 Metric space and Baire Category theorem 15 Continuous functions and Distance in metric space 151 Hausdorff distance and Gromov-Hausdorff distance 152 Invariant of domain 2 Lebesgue measure 21 Exterior measure 22 Measure 23 Borel sets and Measurable sets 24 Linear transformation of measurable sets 25 Sets of positive measure 3 Measurable functions 31 Measurable functions 32 Simple functions 33 Littlewood’s Three principles 4 Lebesgue’s integration theory 41 Integration 42 Interchanging limits with integrals 43 Lebesgue vs Riemann 44 Fubini’s Theorem 5 Differentiation 51 Monotone functions 52 Fundamental theorem of Calculus I 521 A detour: Bounded variation functions 53 Fundamental theorem of Calculus II 54 Lebesgue Differentiation Theorem 6 Function spaces 61 L P spaces 611 Normed vector space 612 A detour: Convexity and Jensen’s inequality 613 Completeness: Banach space 614 Separability 62 Hilbert space:
