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Transcript of Chapter VIII (to Student)
8/4/2019 Chapter VIII (to Student)
http://slidepdf.com/reader/full/chapter-viii-to-student 1/41
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
8/4/2019 Chapter VIII (to Student)
http://slidepdf.com/reader/full/chapter-viii-to-student 2/41
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
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
http://slidepdf.com/reader/full/chapter-viii-to-student 3/41
Functions
Limit of a Function
Continuity
Functions
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
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
http://slidepdf.com/reader/full/chapter-viii-to-student 4/41
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
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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}.
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Functions
Functionscontinue...
Injection
Surjection
Bijection
Composition of functions
Inverse function
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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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
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
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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 .
NGUYEN CANH Nam Mathematics I - Chapter 8
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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| < .
NGUYEN CANH Nam Mathematics I - Chapter 8
F i Li i fi i i
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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 .
NGUYEN CANH Nam Mathematics I - Chapter 8
F ti Li it t fi it i t
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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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
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
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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 ).
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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Functions
Limit of a Function
Continuity
Limit at a finite point
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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Limit of a Function
Continuity
Limit at infinity
One-sided Limits
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
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions Limit at a finite point
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Limit of a Function
Continuity
Limit at infinity
One-sided Limits
Definitionscontinue..
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| <
Definition
We say that limx →a −
f (x ) = L or that f (x )→
L as x →
a − if
∀ > 0, ∃δ > 0 : 0 < a − x < δ ⇒ |f (x )− L| <
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Li i f F i
Limit at a finite point
Li i i fi 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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Li it f F ti
Limit at a finite point
Li it t i fi it
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Limit at a finite point
Limit at infinity
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Limit at a finite point
Limit at infinity
8/4/2019 Chapter VIII (to Student)
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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 )
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
8/4/2019 Chapter VIII (to Student)
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Limit of a Function
ContinuityDiscontinuity
Uniform 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
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
8/4/2019 Chapter VIII (to Student)
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Limit of a Function
ContinuityDiscontinuity
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Di i i
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Limit of a Function
ContinuityDiscontinuity
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Di ti it
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ContinuityDiscontinuity
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)
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Discontinuity
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ContinuityDiscontinuity
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 )).
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Discontinuity
8/4/2019 Chapter VIII (to Student)
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ContinuityDiscontinuity
Uniform 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 point
Continuity on an interval
Discontinuity
Uniform continuity
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Discontinuity
8/4/2019 Chapter VIII (to Student)
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ContinuityDiscontinuity
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity at a pointContinuity on an interval
Discontinuity
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ContinuityDiscontinuity
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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
C ti it
Continuity at a pointContinuity on an interval
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 ].
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity
Continuity at a pointContinuity on an interval
Discontinuity
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Continuityy
Uniform 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 point
Continuity on an interval
Discontinuity
Uniform continuity
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity
Continuity at a pointContinuity on an interval
Discontinuity
8/4/2019 Chapter VIII (to Student)
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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.
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity
Continuity at a pointContinuity on an interval
Discontinuity
8/4/2019 Chapter VIII (to Student)
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ContinuityUniform 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 point
Continuity on an interval
Discontinuity
Uniform continuity
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity
Continuity at a pointContinuity on an interval
Discontinuity
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ContinuityUniform continuity
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
Continuity at a pointContinuity on an interval
Discontinuity
U if i i
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ContinuityUniform continuity
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
|> .
NGUYEN CANH Nam Mathematics I - Chapter 8
Functions
Limit of a Function
Continuity
Continuity at a pointContinuity on an interval
Discontinuity
U if ti it
8/4/2019 Chapter VIII (to Student)
http://slidepdf.com/reader/full/chapter-viii-to-student 41/41
yUniform continuity
Theorem
TheoremIf f is continuous on [a , b ], then f is uniformly continuous on
[a , b ].
NGUYEN CANH Nam Mathematics I - Chapter 8