Continuity and One-Sided Limits：連續(xù)性和片面的限制

1、Objectives:2.4.1 Describe continuity and be able to distinguish a continuous function from one with discontinuities.Materials:Exploration 2-4a (IRB)Lesson 1Warm Upp. 49-50: Quick Review: Q1-Q101. What is meant by the derivative of a function?2. What is meant by the definite integral of a function?3.

2、 Draw a pair of alternate interior angles.4. What type of function has a graph like the following:5. Sketch the graph of 6. Factor: 7. Evaluate: 8. Evaluate: 5!9. No calculator! Divide 50 by ½ and add 3.Solutions:Q1. Instantaneous rate of changeQ2. Product of x and y, where x varies and y can v

3、ary3.4. Exponential function5. 6. 7. 538. 1209. 103I. Continuity at a Point and on an Open IntervalA function is continuous if there are no holes, steps, or asymptotes. Basically, this means that you can draw the function without lifting your pencil. Some of the most interesting functions with disco

4、ntinuities are “piecewise” functions.Graphing a Piecewise Function ManuallyPiecewise functions have multiple branches at different sections of the domain of the function. To manually graph a piecewise function, graph each piece separately and then erase the sections that are beyond the domain of tha

5、t section.Example 1Manually graph the following piecewise function:Query: Is the function in Example 1 continuous?Solution: No, there is a step discontinuity.Example 2Manually graph the following piecewise function:Query: Is the function in Example 2 continuous?Solution: No, the vertical asymptote i

6、s a discontinuity.Graphing a Piecewise Function with a CalculatorGraph the function from Example 1, ,by entering the following into Y1: ExamplePlot the following function on your calculator: a. Does f(x) have a limit as x approaches 2?No, the function has different values if you approach from the le

7、ft or from the right.b. Is f(x) continuous at x = 2?No, there is a gap.c. Does f(x) have a limit as x approaches 5?d. Is f(x) continuous at x = 5?“Types” of Discontinuities: Holes, Steps, and AsymptotesThere are three basic types of discontinuities: holes, steps, and asymptotes.Figure 1: HoleFigure

8、2: Hole #2Not defined at does not existFigure 3: StepFigure 4: AsymptoteNot defined at and does not existDefinition of ContinuityA function is continuous at c if:.Continuous on an open interval (a, b): A function that is continuous at each point in the interval.Everywhere continuous: A function that

9、 is continuous on the entire real line Removable Discontinuities: discontinuities that can be “removed” by redefining . Holes are removable discontinuities.Nonremovable Discontinuities: discontinuities that cannot be fixed easily.(1) Steps (2) Vertical asymptotesIdentifying DiscontinuitiesThe three

10、types of discontinuities are easily identified by the cartoonish graphs found in the textbook. However, hole and jump discontinuities are invisible on graphing calculators. Therefore, you must be able to identify the discontinuities algebraically.1. Zeros in Denominators of Rational Functions: could

11、 be removable or nonremovable discontinuities.2. Holes in Piecewise Functions: these occur when there is a singular x-value that is not in the domain of the function.3. Steps in Piecewise Functions: these occur when the endpoints of adjacent branches dont match up.4. Toolkit Functions: you must be f

12、amiliar enough with the elementary functions to be able to identify vertical asymptotes, i.e. .5. Plot with a Calculator: for unfamiliar functions, you may be able to identify vertical asymptotes and steps by simply graphing the function. However, remember that holes cannot be seen on the graphs of

13、calculators. Also, you may want to plot the functions in “dot mode” so that vertical asymptotes dont appear to be part of the function.6. TABLE: If you suspect that there is a discontinuity at a particular x-value, check the table on your calculator. If an x-value has an ERROR, then there is a disco

14、ntinuity.Describing ContinuityOn tests and quizzes, you will be asked to “describe the continuity of a function”. The following are possible responses to this question:1. State that the function is “continuous everywhere”2. Identify the type of discontinuity and list the x-value(s) where the discont

15、inuities exist.Activity 1Put students into groups of either 4 or 5. Each student should plot two of the functions in the following set. Then, the students should take turn explaining the continuity of their functions. Students should show their calculators to their partners so that everyone understa

16、nds the continuity description.Graph the following functions. State whether you think each of the functions is continuous for the entire real number line.a. Continuousb. Discontinuity at x = 2.c. Appears to be continuous, but there is a discontinuity at .d. Appears to be continuous, but there is a d

17、iscontinuity at .e. Discontinuity at x = 0.f. Discontinuity at x = 2.g. Discontinuties at h. Continuous everywherei. Vertical asymptote at x = 11j. Discontinuities at every integer (graph in dot mode)k. There is a domain restriction and x = 0 (vertical asymptote). Nonremovable discontinuityl. Contin

18、uous on m. Removable discontinuity at x = -1n. Continuous on o. Continuous on Query 1Which trigonometric functions are continuous on the entire number line?Query 2Define an interval for which the other trigonometric functions are continuous. Sometimes you will be asked to discuss the continuity of a

19、 function for only a finite part of the number line (as opposed to the entire set of real numbers).Example 1Discuss the continuity of the following functions on the given interval:ContinuousHomework: Day 1: p. 50-51: 1-29 oddExit Ticket1. (T/F) If = L, then .False. There could be a step discontinuit

20、y at .Lesson 2Warm UpSketch 5 separate graphs that fit the following descriptions:1. has a value for f(-2) but has no limit as x approaches -22. is continuous at x = 4 and is “smooth” there3. has a value for f(2) and a limit as x approaches, but is not continuous at x = 2.4. the limit of f(x) as x a

21、pproaches 5 is -2, and the value for f(5) is also -2.5. f(3) = 5, but f(x) has no limit as x approaches 3 and no vertical asymptote there.II. One-Sided Limits and Continuity on a Closed IntervalPreviously, we said that if a function approached different values from the left and right at a point x =

22、c, then the limit did not exist there. Now, we will learn about limits that only consider one direction.Limit from the right: limit that only considers values greater than x = c.Limit from the left: limit that only considers values less than x = c.Figure 2: One-sided limitsExample 2Find the limit as

23、 x approaches 5 from the left.0The limit in Example 2 would be written like: Example 3: Step FunctionThe greatest integer function, or “step” function, has a series of gap discontinuities.Figure 3: Step functionFind the limit of the greatest integer function as x approaches 0 from the left and right

24、.Limit from the left: -1Limit from the right: 0In cases such as this, we say that the limit does not exist.*Limits Exist When: Definition of Continuity on a Closed IntervalA function f is continuous on the closed interval if it is:(1) continuous on the open interval and (2) Condition (2) means that

25、at each endpoint, the value of the function is equal to its limit from within the interval. In other words, there can be no jumps or gaps at the endpoints.Continuity on a Closed Interval can include functions such as , which is not defined for x values less than zero. According to the definition, is

26、 continuous on the interval Example 4Discuss the continuity of the function on the closed interval.Example 5Discuss the continuity of the function on the closed interval.Properties of ContinuityIf are continuous at , then the following functions are also continuous at c.1. Scalar multiple: 2. Sum an

27、d difference: 3. Product: 4. Quotient: There are certain functions that are always continuous at every point in their domain.Example 6Determine the domains of the example functions in Table 1. These functions are continuous everywhere in these domains.Table 1: Continuous FunctionsType of FunctionExample DomainPolynomialRationalRadicalTrigonometricNote that these “continuous” functions are not necessarily continuous for all real numbers. Co