Miss Battaglia AB/BC Calculus

1. 1.4 Continuity and One-Sided LimitsObjective: Determine continuity at a point and on an open interval; determine one-sided limits and continuity on a closed interval. Miss Battaglia AB/BC Calculus

2. Continuity • What does it mean to be continuous? Below are three values of x at which the graph of f is NOT continuous At all other points in the interval (a,b), the graph of f is uninterrupted and continuous does not exist f(c) is not defined

3. Definition of Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. f(c) is defined 2. exists 3. 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. A function that is continuous on the entire real line (-∞,∞) is everywhere continuous.

4. Discontinuities • Removable (f can be made continuous by appropriately defining f(c)) & nonremovable. Nonremovable Discontinuity Removable Discontinuity

5. Continuity of a Function • Discuss the continuity of each function a) b) c) d)

6. One-Sided Limits and Continuity on a Closed Interval • Limit from the right • Limit from the left • One-sided limits are useful in taking limits of functions involving radicals (Ex: if n is an even integer)

7. A One-Sided Limit • Find the limit of as x approaches -2 from the right.

8. The Greatest Integer Function • One sided limits can be used to investigate the behavior of step functions. A common type is the greatest integer function defined by = greatest integer n such that n < x • Ex: = 2 and = -3 • Find the limit of the greatest integer function as x approaches 0 from the left and from the right.

9. Theorem 1.10: The Existence of a Limit Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iff and Definition of Continuity on a Closed Interval A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and and The function f is continuous from the right at a and continuous from the left at b.

10. Continuity on a Closed Interval • Discuss the continuity of

11. Theorem 1.11 PROPERTIES OF CONTINUITY If b is a real number and f and g are continuous at x=c, then the following functions are also continuous at c. Scalar multiple: bf Sum or difference: f + g Product: fg Quotient: , if g(c)≠0 By Thm 1.11, it follows that each of the following functions is continuous at every point in its domain.

12. THEOREM 1.12 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. Theorem 1.13 INTERMEDIATE VALUE THEOREM 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 in c in [a,b] such that f(c) = k

13. Intermediate Value Theorem • Consider a person’s height. Suppose a girl is 5ft tall on her thirteenth bday and 5ft 7in tall on her fourteenth bday. For any height, h, between 5ft and 5ft 7in, there must have been a time, t, when her height was exactly h. • The IVT guarantees the existence of at least one number c in the closed interval [a,b]

14. An Application of the IVT • Use the IVT to show that the polynomial function f(x)=x3 + 2x – 1 has a zero in the interval [0,1]

15. Classwork/Homework • AB: Page 78 #27-30, 35-51 odd, 69-75 odd, 78, 79, 83, 91, 99-102 • BC: Page 78 #3-25 every other odd, 31, 33, 34, 35-51 every other odd, 61, 63, 69, 78, 91,99-103