含数学分析和高等代数两门课（Two courses including mathematical analysis and advanced algebra）.doc

含数学分析和高等代数两门课（Two courses including mathematical analysis and advanced algebra） Two courses including mathematical analysis and advanced algebra Mathematical analysis (I) (1) sets and functions Real valued summary, absolute value inequality, interval and neighborhood, bounded set, supremum principle, function concept. (2) sequence limit Sequence. EMBED Equation.3 definition of sequence limit. The properties of convergent series are uniqueness, boundedness, number preserving, inequality property, convergence and rational operation. Subsequence. The condition of the existence of a sequence limit; the monotone finite theorem and the Cauchy convergence principle. EMBED, Equation.3, STOLZ theorem. (3) function limit Function limit concept (EMBED, Equation.3). The limit of an instantaneous function. The definition of EMBED Equation.3 and EMBED Equation.3) the properties of function limit: uniqueness, local boundedness, locality preserving property, inequality property, convergence and rational operation. The conditions for the existence of function limit: the resolution principle and the Cauchy criterion. Two important limits: EMBED, Equation.3 Comparison of infinitesimal quantity and infinite quantity and its order. (4) continuity of function Continuity of a function at a point. Unilateral continuity. Discontinuity point and its classification. A continuous function on an interval. The local properties of continuous functions are boundedness, preservation of numbers, rational operations of continuous functions, and continuity of composite functions. The properties of continuous functions on closed intervals are boundedness, maximum and minimum, betweenness and uniform continuity. Continuity of elementary functions. (5) limit and continuity (Continued) The basic theorem of completeness: interval theorem, Cauchy series convergence criterion, accumulation principle, compactness theorem, finite covering theorem, the fundamental theorem of equivalence of completeness. Descrip