1. Continuity (Informal Definition)
A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting a pencil from the paper.

2. Describe the intervals of continuity for each function.
DETERMINING INTERVALS OF CONTINUTIY
Example 1
Describe the intervals of continuity for each function.
Solution The function is continuous over its entire domain,(– , ).

3. Describe the intervals of continuity for each function.
DETERMINING INTERVALS OF CONTINUTIY
Example 1
Describe the intervals of continuity for each function.
Solution The function has a point of discontinuity at
x = 3. Thus, it is continuous over the intervals ,
(– , 3) and (3, ). 3

4. IDENTITY FUNCTION  (x) = x x y – 2 – 1 1 2
Domain: (– , ) Range: (– , )
IDENTITY FUNCTION  (x) = x y x y
– 2
– 1 1 2 x
(x) = x is increasing on its entire domain, (– , ).
It is continuous on its entire domain.

5. SQUARING FUNCTION  (x) = x2 x y – 2 4 – 1 1
Domain: (– , ) Range: [0, )
SQUARING FUNCTION  (x) = x2 y x y
– 2 4
– 1 1 2 x
(x) = x2 decreases on the interval (– ,0] and increases on the interval [0, ).
It is continuous on its entire domain, (– , ).

6. CUBING FUNCTION  (x) = x3 x y – 2 – 8 – 1 1 2 8
Domain: (– , ) Range: (– , )
CUBING FUNCTION  (x) = x3 y x y
– 2
– 8
– 1 1 2 8 x
(x) = x3 increases on its entire domain, (– ,) .
It is continuous on its entire domain, (– , ).

7. SQUARE ROOT FUNCTION  (x) = x y 1 4 2 9 3 16
Domain: [0, ) Range: [0, )
SQUARE ROOT FUNCTION  (x) = y x y 1 4 2 9 3 16 x
(x) = increases on its entire domain, [0,).
It is continuous on its entire domain, [0, ).

8. CUBE ROOT FUNCTION  (x) = x y – 8 – 2 – 1 1 8 2
Domain: (– , ) Range: (– , )
CUBE ROOT FUNCTION  (x) = y x y
– 8
– 2
– 1 1 8 2 x
(x) = increases on its entire domain, (– , ) .
It is continuous on its entire domain, (– , ) .

9. ABSOLUTE VALUE FUNCTION  (x) = x y – 2 2 – 1 1
Domain: (– , ) Range: [0, )
ABSOLUTE VALUE FUNCTION  (x) = y x y
– 2 2
– 1 1 x
(x) = decreases on the interval (– , 0] and increases on [0, ).
It is continuous on its entire domain, (– , ) .

10. Graph the function. a. Solution GRAPHING PIECEWISE-DEFINED FUNCTIONS
Example 2
Graph the function. a. y 2 4 6
– 2 3 5
(2, 3)
(2, 1)
Solution x

11. Graph the function. b. Solution GRAPHING PIECEWISE-DEFINED FUNCTIONS
Example 2
Graph the function. b. y 2 4 6
– 3 3 5
(1, 5)
Solution x

12. GREATEST INTEGER FUNCTION  (x) =
Domain: (– , ) Range: {y  y is an integer} = {…,– 2, – 1, 0, 1, 2, 3,…}
GREATEST INTEGER FUNCTION  (x) = x y
– 2
– 1.5
– .99
– 1
.001 3
3.99 1 2 3
– 2 4
– 3
– 4
(x) = is constant on the intervals…, [– 2, – 1), [– 1, 0), [0, 1), [1, 2), [2, 3),…
It is discontinuous at all integer values in its domain (– , ).