Continuity and One-Sided Limits

1. Continuity and One-Sided Limits Lesson 1.4

2. Calculus • 1.4 Continuity and 1-sided limits • Student objectives: • Understand & describe & find continuity at a point vs. continuity on an open interval • Find 1-sided limits • Use properties of continuity • Understand & use the Intermediate Value Theorem

3. Question?? • How do I get from point A to point B? School Home

4. Intuitively, a function is continuous at x = c if you can draw it without lifting your pen from the paper. In the diagram below, the function on the left is continuous throughout, but the function on the right is not. It is "discontinuous" at x = c.

5. Formally, a function is continuous at x = c on the open interval (a, b) if the following three conditions are met: f(c) is defined exists

6. 1.4: Examples of Discontinuity • F is continuous at x=c if there are no holes, jumps or gaps at c

7. Graphically: f (x) is continuous at c.

8. 1.4: Removable and Nonremovable discontinuity Removable discontinuity Nonremovable discontinuity Removable discontinuity

9. Discuss the continuity of the following graph.

10. Continuity at a Point • A function can be discontinuous at a point • The function jumps to a different value at a point • The function goes to infinity at one or both sides of the point

11. Discuss the continuity of each of the following functions:

12. The domain of f is all nonzero real numbers. You can conclude that f is continuous at every x-value in its domain. In other words, there is no way to define f(0) so as to make the function continuous at x = 0. Nonremovable Discontinuity

13. Discuss the continuity of each of the following functions:

14. The domain of f is all real numbers except x = 1. Therefore, you can conclude that f is continuous at every x-value in its domain. At x = 1, the function has a removable discontinuity. If f(1) is defined as 2, the “newly defined” function is continuous for all real numbers. Removable Discontinuity

15. Discuss the continuity of each of the following functions:

16. The domain of f(x) is all real numbers. Therefore, you can conclude that the function is continuous on its entire domain. When a function is continuous from (-∞,∞), we say that the function is everywhere continuous. Everywhere Continuous

17. if x ≠ 1 and F(x) = 4 if x = 1 if x ≠ 1 and g(x) = 6 if x = 1 if x ≠ 1 and h(x) = 4 if x = 1 Which of These is Dis/Continuous? • When x = 1 … why or not Are any removable?

18. Continuity Theorem • A function will be continuous at any number x = c for which f(c) is defined, when … • f(x) is a polynomial • f(x) is a power function • f(x) is a rational function • f(x) is a trigonometric function • f(x) is an inverse trigonometric function

19. Theorem 1.11 Properties of Continuity

20. Theorem 1.12 Continuity of a Composite Function

21. has a discontinuity at . Write an extended function that is continuous at . Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

22. Make sure you have all this down A plus sign means from the right and a minus sign means from the left If the limit exists then the limit is the same from the left and from the right. A table can be helpful when you can’t find a general limit.

23. Theorem 1.10 The Existence of a Limit

24. Left & Right Hand Limits

25. Theorem The limit does not exist for this function. The limit exists for this function.

26. The limit can exist even when the function is not defined at a point or has a value different from the limit. The limit is a number

27. Section 1.4: ContinuityRationals • Continuity of a closed interval: • Right Cont. • Left Cont. • Conclude that f is continuous on the closed interval

28. Greatest Integer Function:

29. Greatest Integer Function: The greatest integer function is also called the floor function. The notation for the floor function is: Find its limit as x approaches 0 from the right and left!!!

30. Use piecewise definition for absolute value functions Limit DNE!

32. Properties of Continuous Functions • If f and g are functions, continuous at x = cThen … • is continuous (where s is a constant) • f(x) + g(x) is continuous • is continuous • is continuous • f(g(x)) is continuous

33. One Sided Continuity • A function is continuous from the right at a point x = a if and only if • A function is continuous from the left at a point x = b if and only if a b

34. Continuity on an Interval • The function f is said to be continuous on an open interval (a, b) if • It is continuous at each number/point of the interval • It is said to be continuous on a closed interval [a, b] if • It is continuous at each number/point of the interval and • It is continuous from the right at a and continuous from the left at b

35. Continuity on an Interval • On what intervals are the following functions continuous?

36. Because the function is continuous, it must take on every y value between and . Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and .

37. Locating Roots with Intermediate Value Theorem • Given f (a) and f (b) have opposite sign • One negative, the other positive • Then there must be a root between a and b a b

38. Intermediate value theorem, bounds. Intermediate value theorem: Given a continuous function in the interval [a,b], if f(a) and f(b) are of different signs, then there is at least one zero between a and b. f(3) = -9 f(4) = -7 f(5) = -3 f(6) = 3 There is a zero in the interval [5,6] because there is a sign change, and by intermediate value theorem, a zero must exist in that interval.

39. Discuss the continuity of each.

40. Discuss the continuity of each. If x < 2, the function is a parabola. (continuous) If x > 2, the function is a line. (continuous) To be continuous, the two sides must also meet when x = 2.

41. Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k.

42. Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k. The red graph has 1 c-value. Orange has 1 c-value. Blue has 5 c-values. Translation: If you connect two dots with a continuous function, you must hit every y-value between them at least once.

43. Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: oscillating infinite jump