This document discusses the continuity of functions. It defines a continuous function as one where three conditions are met at a point c: 1) the function value f(c) exists, 2) the limit as x approaches c exists, and 3) the function value and limit value are equal. It then provides several examples of determining if functions are continuous at given points by checking if they meet these three conditions.

1. Continuity of Functions
Grade 11 Basic Calculus

2. Continuity of Functions
• A function 𝑓(𝑥) is said to be continuous at 𝒙 = 𝒄 if the following three
conditions are satisfied:
1. 𝒇 𝒄 exists;
2. 𝐥𝐢𝐦
𝒙→𝒄
𝒇 𝒙 exists; and
3. 𝒇 𝒄 = 𝐥𝐢𝐦
𝒙→𝒄
𝒇(𝒙)
If at least one of these conditions is not met, 𝑓 is said to be discontinuous
at 𝑥 = 𝑐.

3. Continuity of Functions
• Example:
Determine if 𝒇 𝒙 = 𝒙𝟑 + 𝒙𝟐 − 𝟐 is
continuous or not at x = 1.

4. Continuity of Functions
• Example:
Determine if 𝒇 𝒙 =
𝒙𝟐−𝒙−𝟐
𝒙−𝟐
is continuous
or not at x = 0.

5. Continuity of Functions
• Example:
Determine if 𝒇 𝒙 =
𝒙𝟐−𝒙−𝟐
𝒙−𝟐
is continuous
or not at x = 2.

6. Continuity of Functions
• Example:
Determine if
𝑓 𝑥 =
𝑥 + 1 𝑖𝑓 𝑥 < 4,
(𝑥 − 4)2
+3 𝑖𝑓 𝑥 ≥ 4
is continuous or not at 𝑥 = 4.

7. Is the function continuous at 𝑥 = 1?

8. Is the function continuous at 𝑥 = 2?

9. 5
Is the function continuous at 𝑥 = 0?

10. Is the function continuous at 𝑥 = 3?

11. Is the function continuous at 𝑥 = −1?

12. Is the function continuous at 𝑥 = −2?

13. Continuity of Functions
• Example:
Determine if 𝑓 𝑥 = 𝑥4 − 𝑥2 + 1 is
continuous or not at 𝑥 = −1.

15. Continuity of Functions
• Example:
Determine if g 𝑥 =
𝑥2−9
𝑥+3
is continuous
or not at 𝑥 = −3.

16. Continuity of Functions
• Example:
Determine if g 𝑥 =
𝑥−1
𝑥+1
is continuous
or not at 𝑥 = 1.

17. Continuity of Functions
• Example:
Determine if
𝑓 𝑥 =
2𝑥 − 3 𝑖𝑓 𝑥 ≤ 4,
𝑥2
− 𝑥 + 1 𝑖𝑓 𝑥 > 4
is continuous or not at 𝑥 = 4.