Continuous-time random walks and fractional calculus: Theory and applications
Continuity 2.4. Most of the techniques of calculus require that functions be continuous. A function...
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Transcript of Continuity 2.4. Most of the techniques of calculus require that functions be continuous. A function...
Continuity
2.4
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
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
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
2?at x continuous2 x1-2x
2 x1xf(x)function theIs
2at x continuousNot
)2()1
DNEf
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
Types of Discontinuities There are 4 types of discontinuities
Jump Point Essential Removable
The first three are considered non removable
Jump Discontinuity Occurs when the curve breaks at a
particular point and starts somewhere else Right hand limit does not equal left hand limit
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)
Essential Discontinuity Occurs when curve has a vertical
asymptote Limit dne due to asymptote
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.
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.
Continuous Functions in their domains Polynomials Rational f(x)/g(x) if g(x) ≠0 Radical trig functions
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
Find and identify and points of discontinuity
4
5)(
x
xf
Non removable – essential discontinuity
VA at x = 4
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)
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
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)