continuity 2.4. most of the techniques of calculus require that functions be continuous. a function...
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Continuity
2.4
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Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function.
This function has discontinuities at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function
1 2 3 4
1
2
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Show g(x)=x^2 + 1 is continuous at x = 1
2)1()1 g
2)(lim)21
xgx
2)1()(lim)31
gxgx
1)( xatcontinuousisxg
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2?at x continuous2 x1-2x
2 x1xf(x)function theIs
3)2()1 f
existsxf
xf
xf
x
x
x
)(lim
3)(lim
3)(lim)2
2
2
2
3)2()(lim)32
fxfx
2)( xatcontinuousisxf
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2?at x continuous2 x1-2x
2 x1xf(x)function theIs
2at x continuousNot
)2()1
DNEf
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2?at x continuous
2
2
2
12
1
f(x)function theIs 2
x
x
x
x
x
x
4)2()1 f
existsxf
xf
xf
x
x
x
)(lim
3)(lim
3)(lim)2
2
2
2
)2()(lim)32
fxfx
2)( xatousdiscontinuisxf
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Types of Discontinuities There are 4 types of discontinuities
Jump Point Essential Removable
The first three are considered non removable
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Jump Discontinuity Occurs when the curve breaks at a
particular point and starts somewhere else Right hand limit does not equal left hand limit
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Point Discontinuity Occurs when the curve has a “hole”
because the function has a value that is off the curve at that point. Limit of f as x approaches x does not equal f(x)
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Essential Discontinuity Occurs when curve has a vertical
asymptote Limit dne due to asymptote
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Removable Discontinuity Occurs when you have a rational
expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.
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Places to test for continuity Rational Expression
Values that make denominator = 0 Piecewise Functions
Changes in interval Absolute Value Functions
Use piecewise definition and test changes in interval
Step Functions Test jumps from 1 step to next.
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Continuous Functions in their domains Polynomials Rational f(x)/g(x) if g(x) ≠0 Radical trig functions
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Find and identify and points of discontinuity
2 xx
2 x3xf(x) 2
5)2()1 f
dnexf
xf
xf
x
x
x
)(lim
4)(lim
5)(lim)2
2
2
2
Non removable – jump discontinuity
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Find and identify and points of discontinuity
4
5)(
x
xf
Non removable – essential discontinuity
VA at x = 4
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Find and identify and points of discontinuity
56
158)(
2
2
xx
xxxf
15
35)(
xx
xxxf
2 points of disc. (where denominator = 0)
Removable disc. At x = 5
Non removable essential at x = -1 (VA at x = -1)
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Find and identify and points of discontinuity
2 xx
2 x5f(x) 2
5)2()1 f4)(lim
4)(lim
4)(lim)2
2
2
2
xf
xf
xf
x
x
x
)2()(lim)32
fxfx
Non removable point discontinuity
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Find and identify and points of discontinuity
20
1572)(
2
2
xx
xxxf
45
325)(
xx
xxxf
2 points of disc. (where denominator = 0)
Removable disc. At x = 5
Non removable essential at x = -4 (VA at x = -4)