Section 2.4
Continuity

   Math 1a


October 3, 2007
Questions



   True or False
   Right now there are two points on opposite sides of the Earth with
   exactly the same te...
Questions



   True or False
   Right now there are two points on opposite sides of the Earth with
   exactly the same te...
Questions



   True or False
   Right now there are two points on opposite sides of the Earth with
   exactly the same te...
Direct Substitution Property




   Theorem (The Direct Substitution Property)
   If f is a polynomial or a rational funct...
Definition of Continuity




   Definition
   Let f be a function defined near a. We say that f is continuous at
   a if
    ...
Free Theorems




  Theorem
   (a) Any polynomial is continuous everywhere; that is, it is
       continuous on R = (−∞, ∞...
Showing a function is continuous


   Example
                 √
   Let f (x) =       4x + 1. Show that f is continuous at 2.
Showing a function is continuous


   Example
                 √
   Let f (x) =       4x + 1. Show that f is continuous at...
Showing a function is continuous


   Example
                 √
   Let f (x) =       4x + 1. Show that f is continuous at...
The Limit Laws give Continuity Laws



   Theorem
   If f and g are continuous at a and c is a constant, then the
   follo...
Transcendental functions are continuous, too




   Theorem
   The following functions are continuous wherever they are de...
What could go wrong?




  In what ways could a function f fail to be continuous at a point a?
  Look again at the definiti...
Pitfall #1
   Example
   Let
                                x2    if 0 ≤ x ≤ 1
                      f (x) =
            ...
Pitfall #1: The limit does not exist
   Example
   Let
                                  x2    if 0 ≤ x ≤ 1
              ...
Pitfall #2



   Example
   Let
                                  x 2 + 2x + 1
                          f (x) =
         ...
Pitfall #2: The function has no value



   Example
   Let
                                  x 2 + 2x + 1
                ...
Pitfall #3



   Example
   Let
                                  46   if x = 1
                        f (x) =
          ...
Pitfall #3: function value = limit



   Example
   Let
                                   46   if x = 1
                 ...
Special types of discontinuites




   removable discontinuity The limit lim f (x) exists, but f is not
                  ...
Special types of discontinuites




   removable discontinuity The limit lim f (x) exists, but f is not
                  ...
A Big Time Theorem




  Theorem (The Intermediate Value Theorem)
  Suppose that f is continuous on the closed interval [a...
Illustrating the IVT



       f (x)




                       x
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b]



       f (x)




                   ...
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b]



        f (x)


    f (b)




    f ...
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b] and let N
   be any number between f (a...
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b] and let N
   be any number between f (a...
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b] and let N
   be any number between f (a...
Illustrating the IVT
   Suppose that f is continuous on the closed interval [a, b] and let N
   be any number between f (a...
Using the IVT


   Example
   Prove that the square root of two exists.
Using the IVT


   Example
   Prove that the square root of two exists.

   Proof.
   Let f (x) = x 2 , a continuous funct...
Using the IVT


   Example
   Prove that the square root of two exists.

   Proof.
   Let f (x) = x 2 , a continuous funct...
Using the IVT


   Example
   Prove that the square root of two exists.

   Proof.
   Let f (x) = x 2 , a continuous funct...
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
Lesson05    Continuity Slides+Notes
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Lesson05 Continuity Slides+Notes

  1. 1. Section 2.4 Continuity Math 1a October 3, 2007
  2. 2. Questions True or False Right now there are two points on opposite sides of the Earth with exactly the same temperature.
  3. 3. Questions True or False Right now there are two points on opposite sides of the Earth with exactly the same temperature. True or False At one point in your life your height in inches equaled your weight in pounds.
  4. 4. Questions True or False Right now there are two points on opposite sides of the Earth with exactly the same temperature. True or False At one point in your life your height in inches equaled your weight in pounds. True or False At one point in your life you were exactly three feet tall.
  5. 5. Direct Substitution Property Theorem (The Direct Substitution Property) If f is a polynomial or a rational function and a is in the domain of f , then lim f (x) = f (a) x→a
  6. 6. Definition of Continuity Definition Let f be a function defined near a. We say that f is continuous at a if lim f (x) = f (a). x→a
  7. 7. Free Theorems Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (−∞, ∞). (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.
  8. 8. Showing a function is continuous Example √ Let f (x) = 4x + 1. Show that f is continuous at 2.
  9. 9. Showing a function is continuous Example √ Let f (x) = 4x + 1. Show that f is continuous at 2. Solution We have √ lim f (x) = lim 4x + 1 x→a x→2 = lim (4x + 1) x→2 √ = 9 = 3. Each step comes from the limit laws.
  10. 10. Showing a function is continuous Example √ Let f (x) = 4x + 1. Show that f is continuous at 2. Solution We have √ lim f (x) = lim 4x + 1 x→a x→2 = lim (4x + 1) x→2 √ = 9 = 3. Each step comes from the limit laws. In fact, f is continuous on its whole domain, which is − 1 , ∞ . 4
  11. 11. The Limit Laws give Continuity Laws Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f + g 2. f − g 3. cf 4. fg f 5. (if g (a) = 0) g
  12. 12. Transcendental functions are continuous, too Theorem The following functions are continuous wherever they are defined: 1. sin, cos, tan, cot sec, csc 2. x → ax , loga , ln 3. sin−1 , tan−1 , sec−1
  13. 13. What could go wrong? In what ways could a function f fail to be continuous at a point a? Look again at the definition: lim f (x) = f (a) x→a
  14. 14. Pitfall #1 Example Let x2 if 0 ≤ x ≤ 1 f (x) = if 1 < x ≤ 2 2x At which points is f continuous?
  15. 15. Pitfall #1: The limit does not exist Example Let x2 if 0 ≤ x ≤ 1 f (x) = if 1 < x ≤ 2 2x At which points is f continuous? Solution At any point a in [0, 2] besides 1, lim f (x) = f (a) because f is x→a represented by a polynomial near a, and polynomials have the direct substitution property. However, lim f (x) = lim x 2 = 12 = 1 x→1− x→1− lim f (x) = lim+ 2x = 2(1) = 2 x→1+ x→1 So f has no limit at 1. Therefore f is not continuous at 1.
  16. 16. Pitfall #2 Example Let x 2 + 2x + 1 f (x) = x +1 At which points is f continuous?
  17. 17. Pitfall #2: The function has no value Example Let x 2 + 2x + 1 f (x) = x +1 At which points is f continuous? Solution Because f is rational, it is continuous on its whole domain. Note that −1 is not in the domain of f , so f is not continuous there.
  18. 18. Pitfall #3 Example Let 46 if x = 1 f (x) = π if x = 1 At which points is f continuous?
  19. 19. Pitfall #3: function value = limit Example Let 46 if x = 1 f (x) = π if x = 1 At which points is f continuous? Solution f is not continuous at 1 because f (1) = π but lim f (x) = 46. x→1
  20. 20. Special types of discontinuites removable discontinuity The limit lim f (x) exists, but f is not x→a defined at a or its value at a is not equal to the limit at a. jump discontinuity The limits lim f (x) and lim+ f (x) exist, but x→a− x→a are different. f (a) is one of these limits.
  21. 21. Special types of discontinuites removable discontinuity The limit lim f (x) exists, but f is not x→a defined at a or its value at a is not equal to the limit at a. jump discontinuity The limits lim f (x) and lim+ f (x) exist, but x→a− x→a are different. f (a) is one of these limits. The greatest integer function f (x) = [[x]] has jump discontinuities.
  22. 22. A Big Time Theorem Theorem (The Intermediate Value Theorem) Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N.
  23. 23. Illustrating the IVT f (x) x
  24. 24. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) x
  25. 25. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) f (b) f (a) x a b
  26. 26. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). f (x) f (b) N f (a) x a b
  27. 27. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c b
  28. 28. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a b
  29. 29. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c1 c2 c3 b
  30. 30. Using the IVT Example Prove that the square root of two exists.
  31. 31. Using the IVT Example Prove that the square root of two exists. Proof. Let f (x) = x 2 , a continuous function on [1, 2].
  32. 32. Using the IVT Example Prove that the square root of two exists. Proof. Let f (x) = x 2 , a continuous function on [1, 2]. Note f (1) = 1 and f (2) = 4. Since 2 is between 1 and 4, there exists a point c in (1, 2) such that f (c) = c 2 = 2.
  33. 33. Using the IVT Example Prove that the square root of two exists. Proof. Let f (x) = x 2 , a continuous function on [1, 2]. Note f (1) = 1 and f (2) = 4. Since 2 is between 1 and 4, there exists a point c in (1, 2) such that f (c) = c 2 = 2. In fact, we can “narrow in” on the square root of 2 by the method of bisections.

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