Precalculus PreAP/Dual, Revised ©2018 Continuity Section 12.3A Precalculus PreAP/Dual, Revised ©2018 viet.dang@humbleisd.net 2/22/2019 2:25 AM §12.3A: Continuity

Examples of Discontinuous Functions §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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) 2/22/2019 2:25 AM §12.3A: Continuity

Example 1 Identify all discontinuities of the graph below by establishing the undefined values of 𝒇 𝒙 = 𝒙 𝟐 𝟒𝒙+𝟏𝟔 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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) 2/22/2019 2:25 AM §12.3A: Continuity

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 2/22/2019 2:25 AM §12.3A: Continuity

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. 2/22/2019 2:25 AM §12.3A: Continuity

Continuity on a Closed Interval 2/22/2019 2:25 AM §12.3A: Continuity

Example 6 Determine the continuity of 𝒇 𝒙 = 𝟏− 𝒙 𝟐 from [−𝟏, 𝟎] 1 –1 –1 1 2/22/2019 2:25 AM §12.3A: Continuity

Example 7 Determine the continuity of 𝒇 𝒙 = 𝒙+𝟏,𝒙≤𝟎 𝒙 𝟐 +𝟏,𝒙>𝟎 from [−𝟏, 𝟏] 2/22/2019 2:25 AM §12.3A: Continuity

Your Turn Determine the continuity of 𝒇 𝒙 = 𝟓−𝒙,𝒙≤𝟐 𝒙 𝟐 −𝟏,𝒙>𝟐 at 𝒙=𝟐 2/22/2019 2:25 AM §12.3A: Continuity

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