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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 1 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | defined liiii. . . y x exists and liiii. . . . y x yd . A function is continuous if it is continuous at each point in its domain. A function is continuous on the closed interval a b if the function is continuous for each point x G a b and liiii. . . y x y a and limx.b- y x y b . Discontinuous Functions. If a function is not continuous at a point it is called discontinuous at that point. If limx.ịy x exists but is not equal to yd then the function has a removable discontinuity. It is thus named because we could define a continuous function y x limx.e y x for x for x to remove the discontinuity. If both the left and right limit of a function at a point exist but are not equal then the function has a jump discontinuity at that point. If either the left or right limit of a function does not exist then the function is said to have an infinite discontinuity at that point. Example 3.2.1 has a removable discontinuity at x 0. The Heaviside function 0 1 2 1 for x 0 for x 0 for x 0 has a jump discontinuity at x 0. 1 has an infinite discontinuity at x 0. See Figure 3.3. Properties of Continuous Functions. Arithmetic. If u x and v x are continuous at x d then u x v x and u x v x are continuous at x d ỈỆ is continuous at x d if v d 0. Function Composition. If u x is continuous at x d and v x is continuous at x p u d then u v x is continuous at x d. The composition of continuous functions is a continuous function. 54 Figure 3.3 A Removable discontinuity a Jump Discontinuity and an Infinite Discontinuity Boundedness. A function which is continuous on a closed interval is bounded in that closed interval. Nonzero in a Neighborhood. If y 0 then there exists a neighborhood e e e 0 of the point such that y x 0 for x G e e . Intermediate Value Theorem. Let u x be continuous on a b . If u a y u b then there exists G a b such that u y. This is known as the intermediate value theorem. A corollary of this is that if u a and u b are of opposite sign then u x has at least one zero on the .