1. Precalculus PreAP/Dual, Revised ©2018
Continuity
Section 12.3A
Precalculus PreAP/Dual, Revised ©2018
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§12.3A: Continuity

2. Examples of Discontinuous Functions
§12.3A: 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
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§12.3A: 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)
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§12.3A: Continuity

5. Example 1
Identify all discontinuities of the graph below by establishing the undefined values of 𝒇 𝒙 = 𝒙 𝟐 𝟒𝒙+𝟏𝟔
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§12.3A: 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
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§12.3A: 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
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§12.3A: 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
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§12.3A: 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
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§12.3A: 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
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§12.3A: Continuity

11. 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)
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§12.3A: Continuity

12. 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
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§12.3A: Continuity

13. 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.
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§12.3A: Continuity

14. Continuity on a Closed Interval
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§12.3A: Continuity

15. Example 6 Determine the continuity of 𝒇 𝒙 = 𝟏− 𝒙 𝟐 from [−𝟏, 𝟎] 1 –1–1 1
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§12.3A: Continuity

16. Example 7
Determine the continuity of 𝒇 𝒙 = 𝒙+𝟏,𝒙≤𝟎 𝒙 𝟐 +𝟏,𝒙>𝟎 from [−𝟏, 𝟏]
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§12.3A: Continuity

17. Your Turn
Determine the continuity of 𝒇 𝒙 = 𝟓−𝒙,𝒙≤𝟐 𝒙 𝟐 −𝟏,𝒙>𝟐 at 𝒙=𝟐
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§12.3A: Continuity

18. Assignment Page 79 27-53 odd, 61-65 odd 2/22/2019 2:25 AM
§12.3A: Continuity