2.6 - 1 Continuity (Informal Definition) A function is continuous over an interval of its domain if...
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Transcript of 2.6 - 1 Continuity (Informal Definition) A function is continuous over an interval of its domain if...
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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.
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Example 1 DETERMINING INTERVALS OF CONTINUTIY
Describe the intervals of continuity for each function.
Solution The function is continuous over its entire domain,(– , ).
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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
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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
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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
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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
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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
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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
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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
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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
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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
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