Math Analysis I - Limits of Functions Part 4 (Discontinuity & Continuity)

Discontinuity functions

of

1stca.se?elimineblediscontinuity# -

-

Let f be defined on a neighbourhood of x , excluding the point x .

R

fiff, fine l

fix)

fa r )

if E

=

= +

fiff

but let

I fat

F o r

f(I) does n o t exist,

then f is said to have an eliminable (or removable) discontinuity at x and

define the continuous extension (or prolongation) of f at x as:

{e)

f r

{t'" E d o m

×

f- r

( x ) = Life = L ,

t e x t x - E

The choice of terminology is justified by the fact that one can modify the

-

function at x by defining it in x , so that to obtain a continuous map at x .

- - 113403

I I doinj

11×1

e . e

g. = ,

¥ 1 fino

fixt-a-fig, text I C H - t

{1¥, x # o

j a r,

EEFE.is?88usoes: - X = o

/

2. Math Analysis Limits of Functions 1 of 36

Vdcaseijumppointlikin

IR

-

Let f be defined on a neighbourhood of x , excluding possibly the point x .

-

E

fan fief

fiff, I N

F

it +

but both

they end r e e l

e x i s t a r e -

Then the function f is said to have a jump point at x whenever left and right

limits exist at x and are real, but they differ from each other.

-

The function f is said to have a discontinuity of the first kind in x .

-

In such cases there is no continuous extension of f and the gap value of f at x

-

is the difference:

IfIt five 11×1-Giff, text

= + ¥

fix

e . 1

y. =

. I t e

e "

fino fief IN-o

- t o

+ +

e"

fifo. five-fan-1

- o

f i f f TAI-Ling,

then K)

f GAP

= - 1

2. Math Analysis Limits of Functions 2 of 36

3rdcaseltind)

A discontinuity point which is not removable, nor of the first kind is said a

discontinuity of the second kind. It therefore comprehend all the other

situations. ut

Ta l - s i

e . g. has for

l i m i t X # 0

n o

by

since criterion non-existence

of

limits

for

by e x t r a c t i n g

{an}, {Xu}

{bn) subsequences o f

and

t h a t o

such en ' →

bun> o o f l b u )

fig fine,

s i n c e then, t

sin'z

Limo does n o t exist

E-o,

t h e n LA)

i n h a s

kind

discontinuity I

a of

2. Math Analysis Limits of Functions 3 of 36

Continuity functions

of

~

f

The function of the previous examples was qualified as a continuous extension:

jail-fifths-fifths

Such property (and more) are embodied in the definition of continuity.

(continuity)

Definition f: IR

EER,

let I E a

E E ,

f i f

continuous e t I

is

HE, Fo> t h a t

s u c h

o

o , light-fells

I x - E l t

t h e E

d,

E, I

o (m)

m e t r i c d e f i n i t i o n

e q u i v a l e n t l y,

O r

t ' V E FVENE

Uyce) t h a t

such

, (T)

(UAE) V

f topological d e f i n i t i o n

C

The function f is said continuous if it is continuous at every point of its

domain. A sequence is continuous at every point of the domain.

DIFFERENCEFROMTHEDEtFNOFLMT.IE R

loud IE),

n o t ICE)

E E then I , E

Moreover allowed

x - E is lee

Iflx) - l

th)

2- L

t h e l i m i t i n

replace

2. Math Analysis Limits of Functions 4 of 36

!ecifica"y:

- if I E a t

if F o r >

M

by He> t h a t

o , s u c h

o

I x- I l e

the 8

O I

E,

E l l e

I ⇐

then, I C H - f t f i x )

fine

E - f e e l

therefore, I E

i f then

E continuity

× fiff

equivalent fat-fat)

E t o

e t i s of continuous

the

i n case

e s I

e x t e n s i o n

ii) 2E

¢

if I E

said

IT Of

I S O L AT E D

a n POINT

i s

and f I

continuous e t

i s

2. Math Analysis Limits of Functions 5 of 36

( l e s to u d r i g h t t i n u i t y )

Definition

A neighbourhood

defined

function n i g h t

a

o n

LR said CONTINUOUS

E i s

o f O N

E T H E RIGHT

lion I N - I c e ,

x t

A neighbourhood

defined

function left

a

o n

LR said CONTINUOUS

E i s

o f O N

E T H E LEFT

lion fix) text

=

E -

X →

f x x

A function defined in a neighbourhood of is continuous at -

- x

if and only if it is continuous on the right and on the left at .

.

Definition (hiecewisecontity)

I I R

→

f : a, b

A map is piecewise-continuous when it is continuous

everywhere except at a finite number of points, at which the discontinuity

is either removable or a jump.

2. Math Analysis Limits of Functions 6 of 36

R

-2] {o} too)

l - n , [ 2 ,

f :

e . g.. u →

u

-2] {o} too)

E - f - p , [ 2 ,

U u

1-7×21×29

X

II. if#

i * t o

D

- o f

E , 2 E

E

O b u t lol

1)

l - l ,

Indeed E

A =

l-l, f

l ) {o)

E l

n =

zero?

f e t

continuous

is

t o be t h a t

continuous i t be t r u e

m u s t

HE> Fo t h a t

s u c h

i o

o

the Ix-El

E, or

I e

o

I f /

(x) E)

- f t E

E

let end 0=1

take

E > o Ix-Elt x - I - o

then, EE, l

I

X o I

If o - o l - o

-fix) I

(x) =

then I I - o

i s a t

continuous

2. Math Analysis Limits of Functions 7 of 36

E-Miho}

¥

IR,

f: E f (x)

→ =

• ,

study continuity

t h e E

j in

of

JE-ITT, E

every point

since i n

is a c c u m u l a t i o n point f o r E .

o n H E

Then, E

E E , continuous e t

f i s

only

a n d

if i f

lim feel

LAI =

x u e ¥ - ¥ - t e x t

take fine

I t o

f continuous

is

Nevertheless, I h a s a

discontinuity e t z e r o :

lim lion

too, = - a

=

r i o t X - t o -

2. Math Analysis Limits of Functions 8 of 36

{o

l X t o

(x)

f sign x =

= x - o

• X t o

l

-

discontinuity

f

1 h a s a

e t z e r o

1 not

f continuous

is

z e r o

e t fino

I E does exist

fix) not

O E lim F (o)

j i n f - O

X - 7 0

Continuity # not $ contrary of %scontinuity