1. Section 1.4A Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net
Continuity
Section 1.4A
Calculus AP/Dual, Revised ©2017
11/29/ :25 PM
§1.4A: Continuity

2. Examples of Discontinuous Functions
11/29/ :25 PM
§1.4A: Continuity

3. Discontinuity Examples
Removable Discontinuity (You can fill the hole in and continue with the function on the right and left side)
Non-removable discontinuity
Jump
Infinite
Oscillating
11/29/ :25 PM
§1.4A: Continuity

4. Definition of Continuity
A function is continuous at the point 𝒙=𝒄 if and only if:
𝒇(𝒄) is defined
2) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 exists
3) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 = 𝒇(𝒄)
11/29/ :25 PM
§1.4A: Continuity

5. Example 1
Identify all discontinuities of the graph below by establishing the undefined values of 𝒇 𝒙 = 𝒙 𝟐 𝟒𝒙+𝟏𝟔
11/29/ :25 PM
§1.4A: Continuity

6. Example 2
Identify all discontinuities of the graph below by establishing the undefined values of 𝒇 𝒙 = −𝒙−𝟖,𝒙≤−𝟏 − 𝒙 𝟐 −𝟒𝒙−𝟒,𝒙>−𝟏
11/29/ :25 PM
§1.4A: Continuity

7. Your Turn
Identify all discontinuities of the graph below by establishing the undefined values of 𝒇 𝒙 = − 𝒙 𝟐 +𝟐,𝒙≠𝟐 −𝟓,𝒙=𝟐
11/29/ :25 PM
§1.4A: Continuity

8. Piecewise Function Limits
Compare the extremes of the piecewise function
Take the right side of the top extreme and equal it to the left side of the middle equation to establish the gap
Solve for the variable
Repeat process for second variable
11/29/ :25 PM
§1.4A: Continuity

9. Example 3
Solve for the values of 𝒂 and 𝒃 that makes 𝒇 𝒙 continuous for the function, 𝒇 𝒙 = 𝒂𝒙+𝟑 𝒊𝒇 𝒙<𝟓 𝟖 𝒊𝒇 𝟓≤𝒙<𝟔 𝒙 𝟐 +𝒃𝒙+𝟏 𝒊𝒇 𝒙≥𝟔
11/29/ :25 PM
§1.4A: Continuity

10. Example 4
Solve for the values of 𝒂 and 𝒃 that makes 𝒇 𝒙 continuous, 𝒇 𝒙 = 𝒙 𝟐 −𝟒 𝒙−𝟐 𝒊𝒇 𝒙<𝟐 𝒂 𝒙 𝟐 −𝒃𝒙+𝟑 𝒊𝒇 𝟐≤𝒙<𝟑 𝟒𝒙−𝒂+𝒃 𝒊𝒇 𝒙≥𝟑
11/29/ :25 PM
§1.4A: Continuity

11. Your Turn
Solve for the values of 𝒂 and 𝒃 that makes 𝒇 𝒙 continuous, 𝒇 𝒙 = 𝒙 𝟐 −𝟒 𝒙−𝟐 𝒊𝒇 𝒙<𝟐 𝒂 𝒙 𝟐 −𝒃𝒙+𝟑 𝒊𝒇 𝟐≤𝒙<𝟑 𝟐𝒙−𝒂+𝒃 𝒊𝒇 𝒙≥𝟑
11/29/ :25 PM
§1.4A: Continuity

12. Example 5
Given, 𝒇 𝒙 = 𝐥𝐧 𝒙 𝒊𝒇 𝟎<𝒙≤𝟐 𝒙 𝐥𝐧 𝟐 𝒊𝒇 𝟐<𝒙≤𝟒 𝟐 𝐥𝐧 𝒙 𝒊𝒇 𝟒<𝒙≤𝟔 , determine whether 𝐥𝐢𝐦 𝒙→𝟐 𝒇 𝒙 exists
A function is continuous at the point 𝒙=𝒄 if and only if:
𝒇(𝒄) is defined
2) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 exists
3) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 = 𝒇(𝒄)
11/29/ :25 PM
§1.4A: Continuity

13. Your Turn
Given, 𝒇 𝒙 = 𝐥𝐧 𝒙 𝒊𝒇 𝟎<𝒙≤𝟐 𝒙 𝐥𝐧 𝟐 𝒊𝒇 𝟐<𝒙≤𝟒 𝟐 𝐥𝐧 𝒙 𝒊𝒇 𝟒<𝒙≤𝟔 , determine whether 𝐥𝐢𝐦 𝒙→𝟒 𝒇 𝒙 exists
11/29/ :25 PM
§1.4A: Continuity

14. Continuity on a Closed Interval
A function is continuous on the closed interval [𝒂, 𝒃] if it is continuous on the open interval (𝒂, 𝒃) and if 𝐥𝐢𝐦 𝒙→ 𝒂 + 𝒇(𝒂) and 𝐥𝐢𝐦 𝒙→ 𝒃 – 𝒇(𝒃) .
The function 𝒇 is continuous from the right at 𝒂 and continuous from the left at 𝒃.
11/29/ :25 PM
§1.4A: Continuity

15. Continuity on a Closed Interval
11/29/ :25 PM
§1.4A: Continuity

16. Example 6 Determine the continuity of 𝒇 𝒙 = 𝟏− 𝒙 𝟐 from [−𝟏, 𝟎] 1 –1–1 1
11/29/ :25 PM
§1.4A: Continuity

17. Example 7
Determine the continuity of 𝒇 𝒙 = 𝒙+𝟏,𝒙≤𝟎 𝒙 𝟐 +𝟏,𝒙>𝟎 from [–𝟏, 𝟏]
11/29/ :25 PM
§1.4A: Continuity

18. Your Turn Determine the continuity of 𝒇 𝒙 =𝟒− 𝟏𝟔− 𝒙 𝟐 from [−𝟒, 𝟒]
11/29/ :25 PM
§1.4A: Continuity

19. Properties of Continuity
If the functions 𝒇 and 𝒈 are continuous at 𝒙=𝒄, then the following are also continuous at 𝒄 (just at a certain point, not everywhere).
Types:
Scalar Multiple: 𝒃• 𝒇
Sum and Difference: 𝒇± 𝒈
Product: 𝒇𝒈
Quotient: 𝒇 𝒈 if 𝒈(𝒄) ≠ 𝟎
11/29/ :25 PM
§1.4A: Continuity

20. Example 8
If 𝒇 𝒙 =𝟑𝒙 is continuous at 𝒙=𝟐 and 𝒈 𝒙 = 𝟏 𝒙−𝟏 is continuous at 𝒙=𝟐, would 𝒇(𝒙)∙𝒈 𝒙 be continuous at 2? Show all work.
11/29/ :25 PM
§1.4A: Continuity

21. Example 9
If 𝒇 𝒙 =𝟑𝒙 is continuous at 𝒙=𝟐 and 𝒈 𝒙 = 𝟏 𝒙−𝟏 is continuous at 𝒙=𝟐, would 𝒇(𝒙)∙𝒈 𝒙 be continuous at 2? Show all work.
11/29/ :25 PM
§1.4A: Continuity

22. Your Turn
If 𝒇 𝒙 =𝟑𝒙 is continuous at 𝒙=𝟐 and 𝒈 𝒙 = 𝟏 𝒙−𝟏 is discontinuous at 𝒙=𝟏, would 𝒇(𝒙)∙𝒈 𝒙 be continuous at 1? Show all work.
11/29/ :25 PM
§1.4A: Continuity

23. AP Multiple Choice Practice Question 1 (non-calculator)
Consider the function, 𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 𝒙 ,𝒙≠𝟎 𝒌,𝒙=𝟎 . For 𝒇 𝒙 to be continuous at 𝒙=𝟎, the value of 𝒌 must be: (A) 𝟎 (B) 𝟏 (C) 𝟐 (D) Does Not Exist
11/29/ :25 PM
§1.4A: Continuity

24. AP Multiple Choice Practice Question 1 (non-calculator)
Consider the function, 𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 𝒙 ,𝒙≠𝟎 𝒌,𝒙=𝟎 . For 𝒇 𝒙 to be continuous at 𝒙=𝟎, the value of 𝒌 must be:
Vocabulary
Connections and Process
Answer and Justifications
11/29/ :25 PM
§1.4A: Continuity

25. Assignment Page 79 27-53 odd, 61-65 odd 11/29/2018 11:25 PM
§1.4A: Continuity