# MATH 140 Lecture Notes - Lecture 9: Intermediate Value Theorem, Classification Of Discontinuities, Oliver Heaviside

by OC13018

School

McGill UniversityDepartment

Mathematics & Statistics (Sci)Course Code

MATH 140Professor

Ewa DumaLecture

9 MATH140 - Lecture 9 Notes

Continuity:

The limit of a function as x approaches a can be often found by calculating the value of

the function at “a”. Functions with this property are usually called “continuous at a”.

Deﬁnition: A function f is continuous at a number “a” if:

lim (x > a) f(x) = f(a)

The deﬁnition requires three things if f is continuous at a:

•f(a) is deﬁned (that is, a is in the domain of f)

•lim (x > a) f(x) exists

•lim (x > a) f(x) = f(a)

This implies that a continuous function f has the property that a small change in x

produces only a small change in f(x).

If f is deﬁned near a (f is deﬁned on an open interval containing a, except perhaps at a),

we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a

Example:

SECTION 2.5 CONTINUITY

||||

119

CONTINUITY

We noticed in Section 2.3 that the limit of a function as approaches can often be found

simply by calculating the value of the function at . Functions with this property are called

continuous at a. We will see that the mathematical deﬁnition of continuity corresponds

closely with the meaning of the word continuity in everyday language. (A continuous

process is one that takes place gradually, without interruption or abrupt change.)

DEFINITION A function is continuous at a number aif

Notice that Deﬁnition l implicitly requires three things if is continuous at a:

1. is deﬁned (that is, ais in the domain of )

2. exists

3.

The deﬁnition says that is continuous at if approaches as xapproaches a.

Thus a continuous function has the property that a small change in xproduces only a

small change in . In fact, the change in can be kept as small as we please by keep-

ing the change in sufﬁciently small.

If is deﬁned near (in other words, is deﬁned on an open interval containing ,

except perhaps at ), we say that is discontinuous at a(or has a discontinuity at ) if

is not continuous at .

Physical phenomena are usually continuous. For instance, the displacement or velocity

of a vehicle varies continuously with time, as does a person’s height. But discontinuities

do occur in such situations as electric currents. [See Example 6 in Section 2.2, where the

Heaviside function is discontinuous at because does not exist.]

Geometrically, you can think of a function that is continuous at every number in an

interval as a function whose graph has no break in it. The graph can be drawn without

removing your pen from the paper.

EXAMPLE 1 Figure 2 shows the graph of a function f. At which numbers is fdiscontinu-

ous? Why?

SOLUTION It looks as if there is a discontinuity when a!1 because the graph has a break

there. The ofﬁcial reason that fis discontinuous at 1 is that is not deﬁned.

The graph also has a break when , but the reason for the discontinuity is differ-

ent. Here, is deﬁned, but does not exist (because the left and right limits

are different). So fis discontinuous at 3.

What about ? Here, is deﬁned and exists (because the left and

right limits are the same). But

So is discontinuous at 5. M

Now let’s see how to detect discontinuities when a function is deﬁned by a formula.

f

lim

x l 5 f!x""f!5"

limxl5 f!x"f!5"a!5

limxl3 f!x"f!3"

a!3

f!1"

limtl0 H!t"0

af

affa

afaf

x

f!x"f!x"

f

f!a"f!x"af

lim

x l a f!x"!f!a"

lim

x l a f!x"

ff !a"

f

lim

x l a f!x"!f!a"

f

1

a

ax

2.5

NAs illustrated in Figure 1, if is continuous,

then the points on the graph of

approach the point on the graph. So

there is no gap in the curve.

!a, f!a""

f!x, f!x""

f

f(a)

x

0

y

a

y=ƒ

ƒ

approaches

f(a).

As x approaches a,

F IGUR E 1

F IGUR E 2

y

0x

1 2 3 4 5

Based on the graph, we can see that when x = 1 and when

x = 5. When x = 1, we see that it is discontinuous because f(1) is not deﬁned. When x =

5, this is a jump discontinuity because f(5) is different from the left/right hand limits.

The following graphs are more examples of jump/inﬁnite/removable discontinuity.

EXAMPLE 2 Where are each of the following functions discontinuous?

(a) (b)

(c) (d)

SOLUTION

(a) Notice that is not deﬁned, so fis discontinuous at 2. Later we’ll see why is

continuous at all other numbers.

(b) Here is deﬁned but

does not exist. (See Example 8 in Section 2.2.) So fis discontinuous at 0.

(c) Here is deﬁned and

exists. But

so is not continuous at 2.

(d) The greatest integer function has discontinuities at all of the integers

because does not exist if is an integer. (See Example 10 and Exercise 49 in

Section 2.3.) M

Figure 3 shows the graphs of the functions in Example 2. In each case the graph can’t

be drawn without lifting the pen from the paper because a hole or break or jump occurs in

the graph. The kind of discontinuity illustrated in parts (a) and (c) is called removable

because we could remove the discontinuity by redeﬁning at just the single number 2.

[The function is continuous.] The discontinuity in part (b) is called an inﬁ-

nite discontinuity. The discontinuities in part (d) are called jump discontinuities because

the function “jumps” from one value to another.

1 2 3

1

x

y

0

(d) ƒ=[x]

1 2

1

x

y

0

(c) ƒ= if x≠2

1if x=2

≈-x-2

x-2

(b) ƒ= if x≠0

1if

1

x=0

1

x

y

0

1 2 x

y

0

1

(a) ƒ=≈-x-2

x-2

FIGURE 3 Graphs of the functions in Example 2

≈

t!x"!x!1

f

nlimxln #x$

f!x"!#x$

f

lim

xl2 f!x""f!2"

lim

xl2 f!x"!lim

xl2 x2"x"2

x"2!lim

xl2 !x"2"!x!1"

x"2!lim

xl2 !x!1"!3

f!2"!1

lim

xl0 f!x"!lim

xl0 1

x2

f!0"!1

ff !2"

f!x"!#x$f!x"!

%

x2"x"2

x"2if x"2

1 if x!2

f!x"!

%

1

x2if x"0

1 if x!0

f!x"!x2"x"2

x"2

V

120

||||

CHAPTER 2 LIMITS AND DERIVATIVES

Deﬁnition: A function f is continuous from the right at a number a if lim (x > a+) f(x) = f(a)

And f is continuous from the left at a if: lim (x > a-) f(x) = f(a)

Deﬁnition: A function f is continuous on an interval if it is continuous at every number in

the interval. If f is deﬁned only on one side of an endpoint of the interval, we understand

continuous at the endpoint to mean continuous from the right or continuous from the

left.

Example: Show that the function f(x) = 1 - sqrt(1-x^2) is continuous on the interval [-1,1].

Using limit laws, we can see that

SECTION 2.5 CONTINUITY

|| ||

121

DEFINITION A function is continuous from the right at a number aif

and is continuous from the left at aif

EXAMPLE 3 At each integer , the function [see Figure 3(d)] is continuous

from the right but discontinuous from the left because

but M

DEFINITION A function is continuous on an interval if it is continuous at

every number in the interval. (If fis deﬁned only on one side of an endpoint of the

interval, we understand continuous at the endpoint to mean continuous from the

right or continuous from the left.)

EXAMPLE 4 Show that the function is continuous on the

interval

SOLUTION If , then using the Limit Laws, we have

(by Laws 2 and 7)

(by 11)

(by 2, 7, and 9)

Thus, by Deﬁnition l, is continuous at if . Similar calculations show that

and

so is continuous from the right at !1 and continuous from the left at 1. Therefore,

according to Deﬁnition 3, is continuous on .

The graph of is sketched in Figure 4. It is the lower half of the circle

M

Instead of always using Deﬁnitions 1, 2, and 3 to verify the continuity of a function as

we did in Example 4, it is often convenient to use the next theorem, which shows how to

build up complicated continuous functions from simple ones.

x2"!y!1"2!1

f

#!1, 1$f

f

lim

xl1! f!x"!1!f!1"lim

x l !1" f!x"!1!f!!1"

!1#a#1

af

!f!a"

!1!s1!a2

!1!

s

lim

xla !1!x2"

!1!lim

xla s1!x2

lim

xla f!x"!lim

xla

(

1!s1!x2

)

!1#a#1

#!1, 1$.

f!x"!1!s1!x2

f

3

lim

x l n! f!x"!lim

x l n!%x&!n!1"f!n"

lim

x l n" f!x"!lim

x l n" %x&!n!f!n"

f!x"!%x&n

lim

x l a! f!x"!f!a"

f

lim

x l a"

f!x"!f!a"

f

2

1-1

1

x

y

0

ƒ=1-œ„„„„„

1-≈

F I G U R E 4

Therefore, f is continuous from the right at -1 and continuous from the left at 1.

Theorem: If f and g are continuous at a and c is a constant then the following functions

are also continuous at a:

•F + g

•F - g

•Cf

•Fg

•F/g if g(a) ≠ 0

Theorem based on direct substitution property:

•Any polynomial is continuous everywhere; it is continuous on XER

•Any rational function is continuous wherever it is deﬁned; it is continuous on its

domain

The Intermediate Value Theorem:

Suppose that f is continuous on the closed interval [a, b] and let N be any number

between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b) such that

f(c) = N.

which is precisely the statement that the function is continuous at ; that

is, is continuous at . M

EXAMPLE 9 Where are the following functions continuous?

(a) (b)

SOLUTION

(a) We have , where

Now is continuous on since it is a polynomial, and is also continuous everywhere.

Thus is continuous on by Theorem 9.

(b) We know from Theorem 7 that is continuous and

is continuous (because both and are continuous). Therefore, by

Theorem 9, is continuous wherever it is deﬁned. Now is

deﬁned when . So it is undeﬁned when , and this happens

when . Thus Fhas discontinuities when xis an odd multiple of and

is continuous on the intervals between these values (see Figure 7). M

An important property of continuous functions is expressed by the following theorem,

whose proof is found in more advanced books on calculus.

THE INTERMEDIATE VALUE THEOREM Suppose that is continuous on the

closed interval and let be any number between and , where

. Then there exists a number in such that .

The Intermediate Value Theorem states that a continuous function takes on every inter-

mediate value between the function values and . It is illustrated by Figure 8. Note

that the value can be taken on once [as in part (a)] or more than once [as in part (b)].

If we think of a continuous function as a function whose graph has no hole or break,

then it is easy to believe that the Intermediate Value Theorem is true. In geometric terms it

says that if any horizontal line is given between and as in Fig-

ure 9, then the graph of can’t jump over the line. It must intersect somewhere.

It is important that the function in Theorem 10 be continuous. The Intermediate Value

Theorem is not true in general for discontinuous functions (see Exercise 44).

One use of the Intermediate Value Theorem is in locating roots of equations as in the

following example.

f

y!Nf

y!f!b"y!f!a"y!N

(b)

0x

y

f(b)

N

f(a)

a c£ b

y=ƒ

c™c¡

(a)

0x

y

f(b)

N

f(a)

a c b

y=ƒ

FIGURE 8

N

f!b"f!a"

f!c"!N!a, b"cf !a""f!b"

f!b"f!a"N#a, b$

f

10

!

x!"

!

, "3

!

, . . .

cos x!#11 $cos x%0

ln!1$cos x"F!x"!f!t!x""

y!cos xy !1

t!x"!1$cos xf !x"!ln x

!h!f"tf!t

f!x"!sin xandt!x"!x2

h!x"!f!t!x""

F!x"!ln!1$cos x"h!x"!sin!x2"

V

af "tah!x"!f!t!x""

126

||||

CHAPTER 2 LIMITS AND DERIVATIVES

FIGURE 7

y=ln(1+cos x)

2

_6

_10 10

b

0x

y

f(a)

N

f(b)

a

y=ƒ

y=N

FIGURE 9

**Unlock Document**

###### Document Summary

We noticed in section 2. 3 that the limit of a function as approaches can often be found simply by calculating the value of the function at . Functions with this property are called continuous at a. We will see that the mathematical de nition of continuity corresponds closely with the meaning of the word continuity in everyday language. (a continuous process is one that takes place gradually, without interruption or abrupt change. ) Math140 - lecture 9 notes a a x. Continuity: is continuous at a number a if. The limit of a function as x approaches a can be often found by calculating the value of the function at a . Functions with this property are usually called continuous at a . De nition: a function f is continuous at a number a if: Notice that de nition l implicitly requires three things if f a lim (x > a) f(x) = f(a)

## More from OC13018

###### MATH 140 Lecture Notes - Even And Odd Functions, Negative Number, Asteroid Family

Lecture Note

###### MATH 140 Lecture Notes - Lecture 4: Exponential Function, Irrational Number, Rational Number

Lecture Note

###### MATH 140 Lecture Notes - Lecture 2: Algebraic Function, Phenylalanine, Cubic Function

Lecture Note

## Classmates also unlocked

###### MATH 140 Lecture Notes - Even And Odd Functions, Negative Number, Asteroid Family

Lecture Note

###### MATH 140 Lecture Notes - Lecture 4: Exponential Function, Irrational Number, Rational Number

Lecture Note

###### MATH 140 Lecture Notes - Lecture 2: Algebraic Function, Phenylalanine, Cubic Function

Lecture Note

###### MATH 140 Lecture Notes - Lecture 8: Squeeze Theorem, Ator, Special Effect

Lecture Note

###### MATH 140 Lecture Notes - Lecture 11: Difference Quotient, Fax, Hyperbola

Lecture Note

###### MATH 140 Lecture Notes - Lecture 6: Macedonian Denar, Central Angle, The Tangent

Lecture Note

###### MATH 140 Lecture Notes - Lecture 10: Rational Number, Indeterminate Form, Asymptote

Lecture Note

###### MATH 140 Lecture Notes - Lecture 5: Cube Root, Foreign Exchange Market, Farad

Lecture Note

###### MATH 140 Lecture Notes - Lecture 3: Eurovision Song Contest, Farad, Function Composition

Lecture Note

###### MATH 140 Lecture Notes - Lecture 13: Power Rule, Constant Function, Ion

Lecture Note

###### Lecture 7 - 2.2.pdf

Lecture Note

###### MATH 140 Lecture Notes - Lecture 12: Differentiable Function, Farad

Lecture Note

###### MATH140 Final: Math 140 notes

Exam Note

###### Calc Notes final 140.doc

Exam Note

###### MATH 140 Lecture Notes - Lecture 3: Virginia State Route 7, Memory Stick

Lecture Note