1.4 Continuity, One-Sided Limits, and Intermediate Value Theorem

Grand Canyon, ArizonaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

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

jump infinite oscillating

Essential Discontinuities:

Removable Discontinuities:

(You can fill the hole.)

Removing a discontinuity:

3

2

1

1

xf x

x

has a discontinuity at .1x

Write an extended function that is continuous at .1x

3

21

1lim

1x

x

x

2

1

1 1lim 1 1x

x x xx x

1 1 1

2

3

2

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.

1x

Removing a discontinuity:

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.

1x

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

Also: Composites of continuous functions are continuous.

examples: 2siny x cosy x

7

Definition

A function f is continuous at a point x = c if

x clim f (x) exists

2.

f (c) is defined 1.

x clim f (x) f (c)

3.

THIS IS THE DEFINITION OF CONTINUITY

8

Examplef (x) = x – 1 at x = 2.

x 2f (2) 1 lim x 1

c.

x 2lim x 1 1

b.The limit exist!

f (2) =a. 1

Therefore the function is continuous at x = 2.

2

1

9

Example

f (x) = (x2 – 9)/(x + 3) at x = -3

2

x 3

x 9lim f ( 3)

x 3

c.

2

x 3

x 9lim

x 3

b. - 6

The limit exists!

f (-3) = 0/0a. Is undefined!

Therefore the function is not continuous at x = -3.

-3

-6

You can use table on your calculator to verify this.

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Continuity Properties

If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0.

Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero.

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Continuity Summary.

3. We have discontinuity with some functions that have a gap.

Graph on your calculator with a standard window.

Functions have three types of discontinuity. Consider - 2

2

x 4x 5f ( x )

x 2x 15

1. Discontinuity at vertical asymptote.

2. Discontinuity at hole.

Intermediate Value Theorem

If a function is continuous between a and b, then it takes

on every value between and . f a f b

a b

f a

f b

Because the function is continuous, it must take on every y value between and .

f a f b

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Intermediate Value Theorem: Intuition

• Traveling on France’s TGV trains, you reach speed of 280 mi/hr.

• How do you know at some point of train ride you were traveling 100 mi/hr?

• To go from 0 to 280, must have passed through 100 mi/hr since speed of train changed continuously

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Intermediate Value Theorem

• Suppose that f is continuous on the closed interval [a,b]. If L is any real number between f(a) and f(b) then there must be at least one number c on the open interval (a,b) such that f(c) = L.

15

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Limitations of IVT• If d(0) = 100 and d(10) = 35, where t is measured in seconds. • d is a continuous function, the IVT tells you that at some point

between t=0 and t =10, the decibel level reached every value between 35 and 100.

• It does NOT say anything about: • When or how many times (other than at least once) a particular

decibel was attained. • Whether or not decibel levels bigger than 100 or less than 35

were reached.

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The Difference Between VROOOOOOOOM and VROOOOOOOM.

These graphs of PC's noise illustrate that very different behaviors are consistent with the hypothesis that d(t) is continuous and that its values at t=0 and t=10 are 100 and 35 respectively.

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Example 1:

• Sketch a graph to decide if the cosecant function, f(x) = csc (x) is continuous over the domain [-π, π].

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Example 2

• Consider the equation sin x = x – 2 . Use the intermediate Value Theorem to explain why there must be a solution between π/2 and π.

Example 3

• Consider the function ,

• Calculate f(6), f(-5.5), f(0)

• Can you conclude that there must be a zero between f(6) and f(-5.5)?

Example 5: Is any real number exactly one less than its cube?

(Note that this doesn’t ask what the number is, only if it exists.)

3 1x x

30 1x x

3 1f x x x

1 1f 2 5f

Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5.Therefore there must be at least one solution between 1 and 2.

Use your calculator to find an approximate solution.

3solve 1,x x x

F2 1: solve

1.32472

This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition.

Graphing calculators can sometimes make non-continuous functions appear continuous.

Graph: floory x

CATALOG F floor(

Note resolution.

The calculator “connects the dots” which covers up the discontinuities.

Graphing calculators can make non-continuous functions appear continuous.

Graph: floory x

CATALOG F floor(

GRAPH

The open and closed circles do not show, but we can see the discontinuities.

If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph.