Chapter VIII (to Student)

8/4/2019 Chapter VIII (to Student)

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Functions

Limit of a Function

Continuity

Functions and continuity

NGUYEN CANH Nam1

1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics

Hanoi University of [email protected]

HUT - 2010

NGUYEN CANH Nam Mathematics I - Chapter 8

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Functions

Limit of a Function

Continuity

Agenda

1 Functions

Functions

2 Limit of a Function

Limit at a finite pointLimit at infinity

One-sided Limits

3 Continuity

Continuity at a pointContinuity on an interval

Discontinuity

Uniform continuity


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Functions

Limit of a Function

Continuity

Functions

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity


Discontinuity

Uniform continuity


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Functions

Limit of a Function

Continuity

Functions

Definition

Definition

A function if on a set D ⊆ IR into a set S ⊆ IR is mapping fromD to S .

In this definition D = D (f ) is the domain of the function f . The

range R (f ) of f is the subset of S consisting of all values f (x ) of

the function.


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Functions

Limit of a Function

Continuity

Functions

Operations on FunctionsAddition, Subtraction, Multiplication and Division

The sum of two functions f and g , denoted f + g , is defined by:

(f + g )(x ) = f (x ) + g (x )The domain of f + g is the set {x | x ∈ D (f ) and x ∈ D (g )}.

The difference of two functions f and g , denoted f − g , is defined by:

(f − g )(x ) = f (x ) − g (x )The domain of f − g is the set {x | xinD (f ) and x ∈ D (g )}.

The product of two functions f and g , denoted fg , is defined by:

(fg )(x ) = f (x )g (x )The domain of fg is the set {x | x ∈ D (f ) and x ∈ D (g )}.

The division of two functions f and g , denoted f g

, is defined by:

f

g (x ) =

f (x )

g (x )

The domain off

g is the set {x |∈ D (f ) and x ∈ D (g ) with g (x ) = 0}.


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Functions

Limit of a Function

Continuity

Functions

Properties of a function

Definition

Let f : D → IR be a real-valued function. f is said to be:

1. Increasing, if∀

a , b ∈

D (a ≤

b ⇒

f (a )≤

f (b ))

2. Decreasing, if ∀a , b ∈ D (a ≤ b ⇒ f (a ) ≥ f (b ))

3. Strictly increasing, if ∀a , b ∈ D (a < b ⇒ f (a ) < f (b ))

4. Strictly decreasing, if ∀a , b ∈ D (a < b ⇒ f (a ) > f (b ))

5. Monotone, if it is either increasing or decreasing.

6. Strictly monotone, if it is either strictly increasing or strictlydecreasing.


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Functions

Limit of a Function

Continuity

Functions

Properties of a functioncontinue..

Definition

Let f : D → IR be a real-valued function. f is said to be:

7. Bounded above, if its range is bounded above, that is if

∃M ∈ IR : ∀x ∈ D ,

f (x ) ≤ M 8. Bounded below, if its range is bounded below, that is if

∃m ∈ IR : ∀x ∈ D , f (x ) ≥ m

9. Bounded, if it is both bounded above and below.

10. Even, if ∀x ∈ D ,−x ∈ D and f (−x ) = f (x )

11. Odd, if ∀x ∈ D ,−x ∈ D and f (−x ) = −f (x )

12. Periodic, if ∃T = 0 ∈ IR : ∀x ∈ D ,

x + T ∈ D

x − T ∈ D and f (x + T ) = f (x ).

The smallest such T is called the period of the function.


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Functions

Limit of a Function

Continuity

Functions

Functionscontinue...

Injection

Surjection

Bijection

Composition of functions

Inverse function


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Functions

Limit of a Function

Continuity

Limit at a finite point

Limit at infinity

One-sided Limits

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity


Discontinuity

Uniform continuity


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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Definition

Definition

We say that limx →a f (x ) = L or that f (x ) → L as x → a if∀ > 0, ∃δ > 0 : 0 < |x − a | < δ ⇒ |f (x )− L| < .

|x − a | represents how far x is from a . The above statement

says that f (x ) can be made arbitrarily close to L simply by

taking x close enough to a .


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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Example

Example

Prove that limx →2

(x + 5) = 7.

Given > 0, we must prove that there exists δ > 0 such that0 < |x − 2| < δ ⇒ |x + 5− 7| < . Let > 0 be given.

|x + 5− 7| < ⇔ |x − 2| <

Thus we see that δ = will work. Indeed, given > 0, 0 < |x − 2| < δ ⇒ |x + 5− 7| < .


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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limits equal infinityDefinition

Definition

We say that limx →a

f (x ) = ∞ or that f (x ) →∞ as x → a if

∀M > 0, ∃δ > 0 : 0 < |x − a | < δ ⇒ f (x ) > M .


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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limits equal infinityExample

Example

Prove that limx →1

1

(x − 1)2= ∞.

Given M > 0, we must prove that there exists δ > 0 such that

0 < |x − 1| < δ ⇒ 1

(x − 1)2> M .

1

(x

−1)2

> M ⇔ (x − 1)2 <1

M

⇔ −1√M

< x − 1 < 1√M ⇔ |x − 1| < √M

Thus, given M > 0, δ =1√M

will work.


Functions Limit at a finite point

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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limits equal infinitycontinue..

Definition

We say that limx →a

f (x ) = −∞ or that f (x ) → −∞ as x → a if

∀M < 0, ∃δ > 0 : 0 < |x − a | < δ ⇒ f (x ) < M .

Remark

In the last two definition, the vertical line x = a is a vertical

asymptote for the graph of y = f (x ).

Remark

When we say that the limit of a function exists, we mean that it

exists and is finite. When the limit is infinite, it does not exist in

the sense that it is not a number. However, we know what the

function is doing, it is approaching ±∞.NGUYEN CANH Nam Mathematics I - Chapter 8


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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity


Discontinuity

Uniform continuity



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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limit at infinityDefinition

Definition

We say that limx →∞

f (x ) = L or that f (x )→

L as x →∞

if

∀ > 0, ∃w > 0 : x ≥ w ⇒ |f (x )− L| <

|f (x )− L| represents the distance between f (x ) and L. The

above statement simply says that f (x ) can be made as close as

one wants from L, simply by taking x large enough. Graphically,this simply says that the line y = L is a horizontal asymptote for

the graph of y = f (x ).



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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limit at infinityExample

Example

Prove that limx →∞

1

x = 0.

Let > 0 be given. We want to find w > 0 so that

x ≥ w ⇒ |1x − 0| < . As usual, we begin with the inequality we are

trying to prove.

|1x − 0| < ⇒ 1

|x | <

Since we are considering the limit as x →∞, we can restrictourselves to positive values of x . Thus, the above inequality can be

replaced with1

x < which is equivalent to x > 1

. Thus we see that

given > 0, w =1

will work.



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Functions

Limit of a Function

Continuity


Limit at infinity

One-sided Limits

Limit at infinitycontinue...

Definition

We say that limx →∞

f (x ) = ∞ or that f (x ) →∞ as x →∞ if

∀M > 0, ∃w > 0 : x ≥ w ⇒ f (x ) > M .

The above definition says that f (x ) can be made arbitrarily

large, simply by taking x large enough.



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Limit of a Function

Continuity

p

Limit at infinity

One-sided Limits

Limit at infinitycontinue...

Example

Prove that limx →∞

x 2 =

∞.

Given M > 0, we must prove that there exists w > 0 such that

x ≥ w ⇒ x 2 > M . Since we are considering the limit as

x →∞, we can restrict ourselves to x > 0. In this case

x

2

> M ⇔ x >

√M

Thus, we see that given M > 0, w =√

M will work.



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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity


Discontinuity

Uniform continuity



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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Definitions

When we say x → a , we realize that x can approach a from two

sides.

If x approaches a from the right, that if x approaches a and isgreater than a , we write x → a +.

Similarly, if x approaches a from the left, that is if x approaches

a and is less than a , then we write x → a −.

We can rewrite the above definition for one sided limits with

little modifications. We do it for a few of them.



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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Definitionscontinue..

Definition

We say that limx →a +


L as x →

a + if

∀ > 0, ∃δ > 0 : 0 < x − a < δ ⇒ |f (x )− L| <

Definition

We say that limx →a −


L as x →

a − if

∀ > 0, ∃δ > 0 : 0 < a − x < δ ⇒ |f (x )− L| <


Functions

Li i f F i


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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Example

Example

Prove that lim

x →

0+

1

x

=

∞.

Given M > 0, we need to prove that there exists δ > 0 such that

0 < x − 0 < δ ⇒ 1x > M .

1

x > M

⇔x <

1

M

Thus, given M > 0, we see that δ = 1M will work.


Functions

Li it f F ti


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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Theorem

Theorem

The following two conditions are equivalent

(i) limx →a

f (x ) = L

(ii) limx →a +

= limx →a −

= L

Remark

One way to prove that limx →a f (x ) does not exits is to prove that the two one-sided limits are not the same or that at least one of

them does not exist.


Functions

Limit of a Function


Limit at infinity

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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Elementary Theorems

Theorem

If the limit of a function exists, then it is unique.

Theorem

Suppose that f (x ) ≤ g (x ) ≤ h (x ) in a deleted neighborhood of

a and lim

x →

a

f (x ) = lim

x →

a

h (x ) = L then lim

x →

a

g (x ) = L.


Functions

Limit of a Function


Limit at infinity

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Limit of a Function

Continuity

Limit at infinity

One-sided Limits

Operations with Limits

Theorem

Assuming that limx →a

f (x ) and limx →a

g (x ) exist, the following results are true:

1 limx →a

(f (x ) ± g (x )) = limx →a

f (x ) ± limx →a

g (x )

2 limx →a (

f (

x )

g (

x )) =

limx →a

f (

x )

limx →a

g (

x )

3 limx →a

f (x )

g (x )=

limx →a

f (x )

limx →a

g (x )as long as lim

x →a g (x ) = 0

4 limx →a

|f (x )| = | limx →a

f (x )|

5 If f (x ) ≥ 0 then limx →a

f (x ) ≥ 0

6 If f (x ) ≥ g (x ) then limx →a

f (x ) ≥ limx →a

g (x )

7 If f (x ) ≥ 0 then limx →a

f (x ) =

lim

x →a f (x )


Functions

Limit of a Function


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Limit of a Function

ContinuityDiscontinuity

Uniform continuity

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity


Discontinuity

Uniform continuity


Functions

Limit of a Function


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Limit of a Function


Uniform continuity

Definition

Definition

We say that a function f is continuous at an interior point c of its

domain if

limx →c f (x ) = f (c ).

If either lim x → cf (x ) fails to exist, or it exists but it not equal to

f (c ), then we say that f is discontinuous at c .

In graphical term, f is continuous at an interior point c of itsdomain if its graph has no break in at the point (c , f (c )); in other

words, if you can draw the graph through that point without

lifting your pen from the paper.


Functions

Limit of a Function


Di i i

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Limit of a Function


Uniform continuity

Right continuity and left continuity

Definition

We say that f is right continuous at c if limx →c +

f (x ) = f (c )

We say that f is right continuous at c if limx →c − f (x ) = f (c )

Definition

We say that f is continuous at a left end point c of its domain if

it is right continuous there.We say that f is continuous at a right end point c of its domain if

it is left continuous there.


Functions

Limit of a Function


Di ti it

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Uniform continuity

Theorems

Theorem

If the functions f and g are both defined on an interval

containing c, and are both continuous at c, then the following

functions are also continuous at c :

The sum f + g and the difference f − g.

The product fg

The constant multiple kf, where k is any number

The quotient f /g (provided g (c ) = 0 )

The nth rooth f (x )1/n (provided f (c ) > 0 if n is even)


Functions

Limit of a Function


Discontinuity

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Uniform continuity

Theoremscontinue..

Theorem

If f (g (x )) is defined on an interval containing c, and if f is

continuous at L and limx →c

g (x ) = L, then

limx →c

f (g (x )) = f (L) = f ( limx →c

g (x )).

In particular, if g is continuous at c (so L = g (c ) ), then the

composition f ◦ g is continuous at c :

limx →c

f (g (x )) = f (g (c )).


Functions

Limit of a Function


Discontinuity

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Uniform continuity

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity

Continuity at a point

Continuity on an interval

Discontinuity

Uniform continuity


Functions

Limit of a Function


Discontinuity

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Uniform continuity

Definition

Definition

We say that function f is continuous on the interval I if it iscontinuous at each point of I . In particular, we will say that f is a

continuous function if f is continuous at every point of its

domain.


Functions

Limit of a Function


Discontinuity

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Uniform continuity

Theorems

Theorem (Min-max theorem)

If f (x ) is continuous on the closed, finite interval [a , b ], then

there exists numbers x 1 and x 2 in [a , b ] such that for all

x ∈ [a , b ], we have

f (x 1) ≤ f (x ) ≤ f (x 2)

Thus f has the absolute minimum value m = f (x 1), taken on at

the point x 1, and the absolute maximum value M = f (x 2), taken on at the point x 2.


Functions

Limit of a Function

C ti it


Discontinuity

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Continuitysco t u ty

Uniform continuity

Theorems

Theorem (Intermediate-Value theorem)

If f (x ) is continuous on the interval [a , b ] and if s is a number

between f (a ) and f (b ), then there exists a number c ∈ [a , b ]such that f (c ) = s.

In particular, a continuous function defined on a closed interval

takes on all values between its minimum value m and its

maximum value M , so its range is also a closed interval [m , M ].


Functions

Limit of a Function

Continuity


Discontinuity

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Continuityy

Uniform continuity

Agenda

1 Functions

Functions



One-sided Limits

3 Continuity



Discontinuity

Uniform continuity


Functions

Limit of a Function

Continuity


Discontinuity

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ContinuityUniform continuity

Definition

Definition

If a function f that is discontinuous at a point a can be redefined

at that single point so that it becomes continuous there, then

we say that f has a removable discontinuity at a . We also saythat a is a removable discontinuous point.

Example

The function g (x ) =

x if x = 21 if x = 2

has a removable

discontinuity at x = 2. To remove it, redefined g (2) = 2.


Functions

Limit of a Function

Continuity


Discontinuity

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Agenda

1 Functions

Functions



One-sided Limits

3 Continuity



Discontinuity

Uniform continuity


Functions

Limit of a Function

Continuity


Discontinuity

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Definition

Definition

We say that a function f is uniformly continuous on a set I ⊂ IR

if and only if

∀ > 0,

∃δ > 0

∀x , y

∈I ,|x −

y |

< δ⇒ |

f (x )−

f (y )|

< .

Example

Let S = IR and f (x ) = 3x + 7. Then f is uniformly continuous

on S .

Choose > 0. Let δ =

3 . Choose x 0 ∈ S . Choose x ∈ S .Assume |x − x 0| < δ. Then

|f (x )− f (x 0)| = |(3x + 7)− (3x 0 + 7)| = 3|x − x 0| < 3δ =

. NGUYEN CANH Nam Mathematics I - Chapter 8

Functions

Limit of a Function

Continuity


Discontinuity

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Examplecontinue...

Example (A counter example)

The function f (x ) = x 2 is continuous but not uniformly

continuous on IR.

Indeed, let = 1. Let δ > 0 be arbitrary. For n ∈IN let

x n = √n + 1, y n = √n . Then x 2n − y 2n = 1, but

0 < x n − y n =(√

n + 1−√n )(√

n + 1 +√

n )√n + 1 +

√n

=1√

n + 1 +√

n ≤ 1√

n +√

n < δ

if n > (1/2δ)2. Thus ∃ > 0,∀δ > 0,∃x , y : |x − y | < δ and

|x 2

−y 2

|> .


Functions

Limit of a Function

Continuity


Discontinuity

U if ti it

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yUniform continuity

Theorem

TheoremIf f is continuous on [a , b ], then f is uniformly continuous on

[a , b ].


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Documents

Chapter VIII (to Student)