2.6 - 12.6 - 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.

2.6 - 22.6 - 2

Example 1 DETERMINING INTERVALS OF CONTINUTIY

Describe the intervals of continuity for each function.

Solution The function is continuous over its entire domain,(– , ).

2.6 - 32.6 - 3

Example 1 DETERMINING INTERVALS OF CONTINUTIY

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

2.6 - 42.6 - 4

Domain: (– , ) Range: (– , )

IDENTITY FUNCTION (x) = x

x y

– 2 – 2

– 1 – 1

0 0

1 1

2 2

(x) = x is increasing on its entire domain, (– , ).It is continuous on its entire domain.

y

x

2.6 - 52.6 - 5

Domain: (– , ) Range: [0, )

SQUARING FUNCTION (x) = x2

x y

– 2 4

– 1 1

0 0

1 1

2 4(x) = x2 decreases on the interval (– ,0] and increases on the interval [0, ).It is continuous on its entire domain, (– , ).

y

x

2.6 - 62.6 - 6

Domain: (– , ) Range: (– , )

CUBING FUNCTION (x) = x3

x y

– 2 – 8– 1 – 10 0

1 1

2 8

(x) = x3 increases on its entire domain, (– ,) .It is continuous on its entire domain, (– , ).

y

x

2.6 - 72.6 - 7

Domain: [0, ) Range: [0, )

SQUARE ROOT FUNCTION (x) =

x y

0 0

1 14 2

9 3

16 4

(x) = increases on its entire domain, [0,).It is continuous on its entire domain, [0, ).

x

( )x x

x

y

x

2.6 - 82.6 - 8

Domain: (– , ) Range: (– , )

CUBE ROOT FUNCTION (x) =

x y

– 8 – 2

– 1 – 1

0 0

1 1

8 2

(x) = increases on its entire domain, (– , ) .It is continuous on its entire domain, (– , ) .

3 x

3( )x x

3 x

y

x

2.6 - 92.6 - 9

Domain: (– , ) Range: [0, )

ABSOLUTE VALUE FUNCTION (x) =

x y

– 2 2

– 1 1

0 0

1 1

2 2(x) = decreases on the interval (– , 0] and increases on [0, ).It is continuous on its entire domain, (– , ) .

x

( )x x

x

y

x

2.6 - 102.6 - 10

Example 2 GRAPHING PIECEWISE-DEFINED FUNCTIONS

Graph the function.

2 5 if 2x x 1 if 2x x

( )x a.

Solution

2 4 6– 2

3

5

(2, 3)(2, 1)

y

x

2.6 - 112.6 - 11

Example 2 GRAPHING PIECEWISE-DEFINED FUNCTIONS

Graph the function. b.

( )x 2 3 if 1x x 6 if 1x x

2 4 6– 3

3

5(1, 5)

Solution

y

x

Domain: (– , )

Range: {y y is an integer} = {…,– 2, – 1, 0, 1, 2, 3,…}

GREATEST INTEGER FUNCTION (x) =

x y

– 2 – 2

– 1.5 – 2

– .99 – 1

0 0

.001 0

3 3

3.99 3

(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 (– , ).

x

x

1 2 3

1

2

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

2.6 - 16