1. Continuity of Function at a Number
A function y = f(x) is continuous at a number a
if and only if the following are satisfied:
If one or more of these three conditions are not satisfied, then the function f(x) is said to be discontinuous at a.

2. equal f is continuous at x = 1.
Graphically, it means there is no break in the graph of f(x) at x = 1.

4. h is not continuous at x = 1.
Not equal
h is not continuous at x = 1.
Graphically, it means there is a break ( jump) in the graph of h(x) at x = 1.

6. f is not continuous at x = 2.
Graphically, it means there is a break (hole) in the graph of f(x) at x = 2.

8. h is not continuous at x = -2.
Graphically, it means there is a break in the graph of h(x) at x = -2 (vertical asymptote).

9. x = -2 (vertical asymptote) y = 2 is a horizontal asymptote
- 4
- 3
- 1
y = h(x) 7 12
- 8
(0, -3) , (3, 0)
vertical asymptote
x = -2
y = 2 is a horizontal asymptote 9

10. Types of Discontinuity
Removable discontinuity – discontinuity where
Graphically, it means there is a break (hole) in the graph of f(x) at x = a and this hole can be patched by redefining f(x) to be equal to its limit L when x = a.
Redefine

12. f has removable discontinuity at x = 2.
Redefine

14. Types of Discontinuity
b) Essential discontinuity – discontinuity where
(i) Jump discontinuity – essential discontinuity where
Graphically, it means there is a break (jump) in the graph of f(x) at x = a. The graph approaches two different points as x approaches a from the left and right.

15. Not equal
h has a jump essential discontinuity at x = 1.

17. Types of Discontinuity
b) Essential discontinuity – discontinuity where
(ii) Infinite discontinuity – essential discontinuity where
Graphically, it means there is a break in the graph of f(x) at x = a where x = a is a vertical asymptote.

18. h has an infinite essential discontinuity at x = -2.

19. x = -2 (vertical asymptote) y = 2 is a horizontal asymptote
- 4
- 3
- 1
y = h(x) 7 12
- 8
(0, -3) , (3, 0)
vertical asymptote
x = -2
y = 2 is a horizontal asymptote 19