tailieunhanh - Handbook of mathematics for engineers and scienteists part 41

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 41', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 248 Limits and Derivatives 3. Let f x be a function on a finite segment a b satisfying the Lipschitz condition f X1 - f X2 L X1 - X2I for any x1 and X2 in a b where L is a constant. Then f x has bounded variation and b V f x L b - a . a 4. Let f x be a function on a finite segment a b with a bounded derivative f x L b where L const. Then f x is of bounded variation and V f x L b - a . 5. Let f x be a function on a b or a to and suppose that f x can be represented as an integral with variable upper limit f x c i p t dt a where p t is an absolutely continuous function on the interval under consideration. Then f x has bounded variation and A fb V f x J x dx. a Corollary. Suppose that p t on a finite segment a b or a to is integrable but not absolutely integrable. Then the total variation of f x is infinite. . Properties of functions of bounded variation. Here all functions are considered on a finite segment a b . 1. Any function of bounded variation is bounded. 2. The sum difference or product of finitely many functions of bounded variation is a function of bounded variation. 3. Let f x and g x be two functions of bounded variation and g x K 0. Then the ratio f x g x is a function of bounded variation. 4. Let a c b. If f x has bounded variation on the segment a b then it has bounded variation on each segment a c and c b and the converse statement is true. In this case the following additivity condition holds b c b V f x V f x V f x 5. Let f x be a function of bounded variation of the segment a b . Then for a x b the variation of f x with variable upper limit x F x V f x a is a monotonically increasing bounded function of x. 6. Any function f x of bounded variation on the segment a b has a left-hand limit lim f x and a right-hand limit lim f x at any point x0 e a b . . Basic Concepts of Mathematical Analysis 249 . Criteria for functions to have bounded variation. 1. A function f x has bounded variation on a finite segment a b if and only if there is a