 Ch 1. Limits Multimedia Engineering Math Limit of aSequence Limit of aFunction LimitLaws Continuity RateChange
 Chapter 1. Limits 2. Derivatives I 3. Derivatives II 4. Mean Value 5. Curve Sketching 6. Integrals 7. Inverse Functions 8. Integration Tech. 9. Integrate App. 10. Parametric Eqs. 11. Polar Coord. 12. Series Appendix Basic Math Units Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Hengzhong Wen Chean Chin Ngo Meirong Huang Kurt Gramoll ©Kurt Gramoll MATHEMATICS - THEORY In this section, the concept of continuous and discontinuous is introduced. Various theorems related to continuous function will be discussed. Continuous Continuous Function Example If then function f(x) is continuous at a. A continuous function implies that: f(a) is defined in the domain of f(x). The limit of f(x) exists at point a. The limit of f(x) at point a equals the value of function f(x) at point a.  Discontinuous Function Example Discontinuous If f(x) is not continuous at point a, then f(x) is discontinuous at a. In other words, f(x) has a discontinuity at point a. Theorems about Continuous In this segment, some theorems are introduced to justify whether a function is continuous or not. A function f(x) is continuous from the left at point a, if A function f(x) is continuous from the right at point a, if Assume that f(x) and g(x) are continuous at point a, then: The constant function h(x) = c is continuous for all x at every a. cf(x) is continuous at a for any constant c. f(x) + g(x) is continuous at a. f(x) - g(x) is continuous at a. f(x)g(x) is continuous at a. f(x)/g(x) is continuous at a if g(a) ≠ 0. f1/n is continuous at a if f(a)1/n exists. Every polynomial function is continuous. Every rational function is continuous where f(x) and g(x) are polynomial functions. g(x) ≠ 0 If f(x) is continuous at b and then If g(x) is continuous at a and f(x) is continuous at g(a), then is continuous at a. In other words, a continuous function of a continuous function is a continuous function. Intermediate Value Theorem Explanation of Intermediate Value Theorem< In order to justify whether a value will fall in a specific range or not, intermediate value theorem is introduced. Assume: f(x) is continuous on the interval [a, b] f(a) ≠ f(b) N is any number between f(a) and f(b) then there exists a point c in [a, b] such that f(c) = N. Corollary of Intermediate Value Theorem f(x) = 0 has 5 roots in [a, b] Corollary of Intermediate Value Theorem is a theorem used to verify whether a function has at lease a root or not. Assume: f(x) is continuous on the [a,b] f(a) and f(b) has opposite signs then the equation f(x) = 0 has at least one root in the open interval (a, b). Extreme Value Theorem Continuous function with minimum and maximum value This is a theorem that can be used for one variable. If f(x) is continuous on [a, b], then f(x) takes on a least value m and a greatest value M on the interval. Discontinuous function without maximum value Theorem Theorem example If f(x) is continuous at point c and      f(c) > 0 then there is a positive number t such that, whenever      c - t < x < c + t then      f(x) > 0.