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)