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Continuity & One-Sided Limits. Section 1.4. After this lesson, you will be able to:. determine continuity at a point and continuity on an open interval determine one-sided limits and continuity on a closed interval understand and use the Intermediate Value Theorem. f. c.

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## Continuity & One-Sided Limits

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**Continuity & One-Sided Limits**Section 1.4**After this lesson, you will be able to:**• determine continuity at a point and continuity on an open interval • determine one-sided limits and continuity on a closed interval • understand and use the Intermediate Value Theorem**f**c Continuous Function Continuity at a Point fis continuous at c if the following three conditions are satisfied: f(c) 1) f(c) is defined Muy importante! Be able to recite by heart.**Some examples of functions that are NOT continuous at c.**1)f is not continuous at c because**Some examples of functions that are NOT continuous at c.**2)f is not continuous at c because**Some examples of functions that are NOT continuous at c.**3)f is not continuous at c because**f**f(c) a c b Continuity on an Open Interval A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. Continuous Function on the open interval (a, b). A function that is continuous over the set of real numbers is called everywhere continuous.**Discontinuity**A function that is defined on the interval (a, b) (except possibly at c), and is not continuous at c is said to have a ___________________ at c. • There are two types of discontinuities: • Removable: If f can be made continuous by defining or redefining f(c)…this type would appear as a _________ in the graph. • Non-removable: f cannot be made continuous by changing only f(c)...e.g. a vertical asymptote at x = c.**Removable Discontinuity**f f(c) Removable discontinuity at x=c, since f can be made continuous by just redefining f(c). c Discontinuity at x = c**Non-removable Discontinuity**This function has a non-removable discontinuity at x = c. If only f(c) was redefined, the function would still be discontinuous. f f(c) c**Continuity**Consider the graphs of the following functions and discuss the continuity of each.**One-sided Limits**Limit from the right(right-hand limit) Limit from the left(left-hand limit)**One-sided Limits***One-sided limits are great for radical functions.* Example: We can’t take the limit of this function as x approaches 0 from the left side since negative numbers are not in the domain. We can only take the right limit.**One-sided Limits**Example: Graph the function on the calculator. Then, determine the limit graphically. What is the domain of this function? ___________________ *One-sided limits are also great for functions with a closed interval as the domain.***Example 3**is called the ______________ __________________ ______________.**Example: Find the one-sided limits at the endpoints of the**function, At –2, we can only take a right limit: 2 -2 At 2, we can only take a left limit: Continuity on a Closed Interval A function is continuous on a closed interval if it is continuous everywhere inside the interval and has one-sided continuity at the endpoints.**Examples**Discuss the continuity of the function on the closed interval.**Properties of Continuity**If k is a real number and f and g are continuous at x = c, then f + g, f-g, f•g, kf, and f/g(provided g(c) 0) are also continuous. The following types of functions are continuous at every point within their domain: • Polynomial Functions • Rational Functions • Radical Functions • Trig Functions**We can also say,**Continuity of a Composite Function If g is continuous at c and f is continuous at g(c), then the composite function given by is continuous at c.**Examples 1 & 2**Describe the intervals on which each function is continuous. Verify graphically.**Examples 3 & 4**Describe the intervals on which each function is continuous. Verify graphically.**Example 5**Describe the intervals on which each function is continuous. Verify graphically.**Example 6**Find the constant a such that f(x) is continuous on .**This theorem doesn’t tell you the value of c, but just**tells you one exists. This is an existence theorem. f(b) k f(a) a c b f(c) = k Intermediate Value Theorem (IVT)“It ain’t a sandwich unless there’s something between the bread.” If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k. A continuous function takes on all values between any two points that it assumes.**Weight (lbs)**Day IVT Example Consider this example: A baby weighs 7.3 lbs on day 1 and weighs 8.9 lbs on day 24. There has to be a time between day 1 & 24 when the baby weighed exactly 8.0 lbs.**Finding Zeros**So how do we use the IVT? We’ll use it primarily to locate zeros of a function that is continuous on an interval.**Finding Zeros**Use the Intermediate Value Theorem to show that has a zero in the interval [0, 1]. Then use your calculator to find the zero accurate to four decimal places.**Homework**Section 1.4:page 78 #1-17 odd, 25, 29 – 37 odd, 69, 71, 77, 79 Don’t Skip!!!!

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