DEFINITION Continuity at a Point f ( x ) is defined on an open interval containing

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31

2.3 Quiz

lim ?4x

x

x x

lim / lim / lim , lim 0

x c x c x c x cf x g x f x g x g x

1 1

5 5

DEFINITION Continuity at a Point f (x) is defined on an open interval containing x = c. If , then f is continuous at x = c .

If the limit does not exist, or if it exists but is not equal to f (c), we say that f has a discontinuity at x = c.

( )a b

Three conditions must hold:

For a function to be continuous at a point...

1. is defined

2. lim exists

3. limx c

x c

f c

f x

f x f c

is continuous at f x c

A function f (x) may be continuous at some points and discontinuous at others. If f (x) is continuous at all points in an interval I, then f (x) is said to be continuous on I. If I is an interval [a, b] or [a, b) that includes a as a left endpoint, we require that

lim .x a

f x f a

Similarly, we require that

if I includes b as a right endpoint. If f (x) is continuous at all points in its domain, then f (x) is simply called continuous.

lim ,x b

f x f b

( )a b

is the open interval , ,

& is continuous on .

I a b

f I

[ ]

limx cf x

exists but is not equal to f (c)

F has a removable discontinuity at x = c.

, 25, 2{ f x xx

f x

A “worse” type of discontinuity is a jump discontinuity, which occurs if the one-sided limits and exist but are not equal. Below are two functions with jump discontinuities at c = 2. Unlike the removable case, we cannot make f (x) continuous by redefining f (c).

limx c

f x

limx c

f x

DEFINITION One-Sided Continuity A function f (x) is called:

•Left-continuous at x = c if •Right-continuous at x = c if

limx c

f x f c

limx c

f x f c

Piecewise-Defined Function Discuss the continuity of

At x = 1, the one-sided limits exist but are not equal:

has a jump discontinuity and is right-continuous at 1.F x x

At x = 3, the left- and right-hand limits exist and both are equal to F (3), so F (x) is continuous at x = 3:

f (x) has an infinite discontinuity at x = c if one or both of the one-sided limits is infinite. Notice that x = 2 does not belong to the domain of the function in cases (A) and (B).

Some functions have more “severe” types of discontinuity. For example, oscillates infinitely often between +1

and −1 as x → 0. Neither the left- nor the right-hand limit exists at x = 0, so this discontinuity is not a jump discontinuity.

1sinf xx

It is easy to evaluate a limit when the function in question is known to be continuous.

limx cf x f c

.

3

lim sin3

2x

x

1

3 1/

6

3im

2

1l

5

x

x x

End Day 1 InstructionBegin

.

Building Continuous Functions

THEOREM 1 Basic Laws of Continuity If f (x) and g (x) are continuous at x = c, then the following functions are also continuous at x = c:

(iv) f (x)/g (x) if g (c) 0

(i) f (x) + g (x) and f (x) – g (x)

(ii) kf(x) for any constant k

(iii) f (x) g (x)

.

Building Continuous Functions

THEOREM 2 Continuity of Polynomial and Rational Functions Let P(x) and Q(x) be polynomials. Then:

(i) P(x) is continuous on the real line.

(ii) P(x)/Q(x) is continuous on its domain.

Building Continuous FunctionsTHEOREM 3 Continuity of Some Basic Functions

(iv) is continuous for (for ).

(i) is continuous on its domain for n a natural #.

(ii) are continuous on the real line.

(iii) is continuous on the real line (for ).

1/ ny x

sin and cosy x y x xy b

logby x 0x 0, 1b b

0, 1b b

1/The domain of is the real line if is odd

and [0, ) if is even.

ny x n

n

x 2x

y 0

1

2

3

1

2

4

8

The base of a logarithmic

function has to be positive.

.

As the graphs suggest, these functions are continuous on their domains.

Because sin x and cos x are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.

(i.e.) there are no jump discontinuities

.

As the graphs suggest, these functions are continuous on their domains.

Because sin x and cos x are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.

sintan

cos

xx

x

tanf x x

Building Continuous Functions

THEOREM 4 Continuity of Composite Functions

If is continuous at , and is continuous

at , then the composite funct

is continuous

i

at

n

o

.

g x c f

x g

F x f g x c

c

x

1/32Is 9 continuous?F x x

is the composite of two continuous functionsF

Yes

Building Continuous Functions

THEOREM 4 Continuity of Composite Functions

If is continuous at , and is continuous

at , then the composite funct

is continuous

i

at

n

o

.

g x c f

x g

F x f g x c

c

x

1Is cos continuous?F x x

1 is continuous 0.g x x x

cos is continuous .f x x

, but is continuous

at all points in its domain.

No F

Building Continuous Functions

THEOREM 4 Continuity of Composite Functions

If is continuous at , and is continuous

at , then the composite funct

is continuous

i

at

n

o

.

g x c f

x g

F x f g x c

c

x

sinIs 2 continuous?xF x

is the composite of two continuous functionsF

Yes

Building Continuous Functions

THEOREM 4 Continuity of Composite Functions

If is continuous at , and is continuous

at , then the composite funct

is continuous

i

at

n

o

.

g x c f

x g

F x f g x c

c

x

2 cos 2 9sin

8

xxF x

x

A function constructed of basic functions

using operatio

Elementary Function

Basic Func

ns and compositions.

A function that's continuous on its do

ti

m n

on

ai .

Continuous : 8x x

2 1.7 1

The Greatest Integer Function

, where is the unique integer such that

1.

x n n

n x n

1.7 ? 2

f x x

2

limx

x

We cannot use susbstitution because

is not continuous.f x x

DNE

Complete