Page 1 I It 2011 by JBn.iC~ L. £PSICln 2.5 ConllnuHY

Continuity (Section 2.5)

f(x) is continuous at a number a if lim f(x) = f(a) X->Q

y (X) EXAMPLE 1

Where is the graph discontinuous?

+

t

')<---2)0 31

- & 5 X coM-a~ ~

f(x) is continuous from the right at a number a if

lim f(x) = f(a) x-a+

f(x) is continuous from the left f( at .' a number a If )x:~ x) = f(a)

'50 oJ 'X '" - S c.rnt- fron1 n2f'..-t ~ 4, Bnm l<?ft

I -ak "/.-- S C,.6Y\t ~ n3't

PaGe 2 1 {j 20 11 by Janice L. Epslcin 2.5 Conli.nui~'

EXAMPLE 2 Explain why each function is discontinuous at the given point.

x2 -1 C( )a)a=-1, f(x)=-- ~ -l D

x +1 ""at

2 X -2x-8 'fIX:;Z: 4

b) a=4, f(x)= x-4 ,j-t(L\) ~ 3 3 If x = 4

\Im ~)(_~)(~+-2-) -:;:;. f..c; f f (1+) )(-"4 (j-t.f) 0 NUl UNT.

A function f(x) is continuous on an interval if it is continuous at

every number in the interval. At the endpoints it is understood that the function is left or right continuous,

Continuity Rules Iffand g are continuous at a and c is a constant, then the following functions are also continuous at a

f±g cf fg f (provided that g(a) :;Z: 0) g

A polynomial is continuous on IR = (-00,00) A rational function is continuous on its domain

Ifn is a positive even integer, f(x) = ~ is continuous on [0,00) If n is a positive odd integer, f(x) is continuous on (-00,00)

Page::; I t· 2lJ 11 b~ Ja ni ce L. EpSICll1 2.5 Conlinu;~

EXAMPLE 3 Find where the' fu .gllve~ nctIOns are discontinuous

x+ Ifx<-1

a) g(x)= 123x if-~<x < 1 2x + 1 ifx > 1

\11'1\ _~i)=(:L"i-i)...\ '" - \ \ I\('() '" ex):: 3(-\T--3 '1-'1- \ )l-7-\tO

'\ (.OY\.+- @) '/..~ - \ \ 11'(\ «.ty.~.,. 3(\) "" 3> I WY'I ~~)" 2LI) ~ \ ='3 y..~ \... u . y."i \r

~ l I)=- '3 ::.) 0.n1\ \;. iIO 'J(~ \ c;loMou("

b) h(t)=~4t2 -1 frvl

:t/Z­lvl 4t

(_cP) -'/,,) V C.'

~ ifx:S;O

h(x)= 1 ifO<x<1

Fx ifx 2: 1I 'O~'(\ Of(

Jh~ hCfC')-- 0 ) JL\YV\ 'n(AJ=- \ 'J, "7 0- '1..-70"r

::::) nor (J)flL

J))v"f\ h ( '}-) :::: \ , ~m ~ (,0 -c;. \

n-y r V!~\\-" '" l\) ~ \ -:;) t.O'f't

Page 4 I t. 201 J by Janice L. Epstein 2. 5 COnlinuiry

EXAMPLE 4 Find the value or values of e that make1 continuous on (- (X), (X))

I(X)=jx 2

_e 2

~fx< 4 5{4)=-c.<.."l)i-ZO ~~O~4G ex + 20 lfx 2: 4

-.O..AY'(\_ ~ lJr- 47._c..'-= \b-C"Z.1 ~e ~& 't-7~ jj..,J r­...Q;.'fY'l R()(') -::. 9..0"'-,~ ~~~~ ~~-\la - C. 'l. :::: ~O +-~ ~) C. +4c.. +--( - 0

C+2..)~ :: 0 -=-'? c.::. - 2 If g is continuous at a and/is continuous at g (a) then

(log )(x) = I(g( x )) is continuous at a .

EXAMPLE 5 Find where the given functions are discontinuous

a) F(x) = 2x + .J25 - x 2

lJz..-\- ~('J,) "" ,"-5 - ,,/-1-- (a ~Jn~).fl ~., ==- fiX LCOVLt:-) ='7 Ffd()()) l<l amt

tr'Y'! olo'rv'OJ..Y' - 5 -£ X ~S

b) g(t) = 1 WAt flY\ clorvw h ~ t + ~ U-J... cA!Y'd t 7/ Z.

c) G(t) = Ix - x21 0JrY-t C'JY\ ~Yl i?-.

d) H (x) = t-2 to'I\\- ()Y\ 0\.0 r"NU-V\ 5+x

~~/6'0f'Jtif . \ ~~ ~ 4' ~ -.., ~ ­<f)-s <;) 2­

( _ (0) ~S ') lJ [2- I 00 )

p" c5 1r: lOll bv Janice L. EpSlein 1.5 Con"nu l ~-e..xJ.. A removable discontinuity at a is a discontinuity that can be removed by redefining the function at a.

EXAMPLE 6 Which of the following functionsfhas a removable discontinuity at a? If it is removable, redefine f so that the function is continuous on ~.

a)

b) a=7, f(x) = x-7Ix- 71

~

. ,_;;'-!- ST (y..j -::- ~~~- ::. \ t \Yv1f Vl

JAW) fo~O -=.. -r(')t--,) ::. -\ 1(1o'\ll) I" 't. 77- '1--, ')3

c) a = -4, f(x) = x + 64 =- ('JL¥Q()L~L}x ~, \~') x + 4 (1OrL.\-) .

g )L-:. -l..\- ) 'Y."'"- til)( +-) ~ ==- 48 CZo d-t..tJ\ f\.t f (-1-\) ~ 4~

d) a=9, f(x)=3-Fx ;: C3-~ 9-x (3-W)C3~)

~ ')L~q I~=r-~;:: y(p 1>0 ~ s:-(q)~1\0

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Intermediate Value Theorem Supposef is continuous on the closed interval [a, bland let N be

any number strictly between f(a) and feb). Then there exists a number c in (a,b) such that fCc) = N .

EXAMPLE 7 Use the Intennediate Value Theorem to show there is a root of the given equation in the given interval.

a) x 5 -2x4 -x-3 = O, (2,3)

-1-(2..) :::.-'5 ~ ft~J= 75 ~cJl. - S< 0 <.. 15

f-(c') -= 0 +err ~<.C <. 3

b) x 2 =.Jx +1, (1, 2)

f('Y-) =- X2.- {ITil. f(\)-=- \-~(O fez.) ::. ~ - ,13 >0

EXAMPLE 8 Use the Intennediate Value Theorem to show that there is a positive number c such that c2 = 2.

1'Y-a--\- ~ {2 ~~~ ~ , ern (l,2) fC.")[):;:.)(2- VJ c®t ~(\ ') -:::. \ '2-~ \

~ (~) ~ :l?'-::: ~

~Vf\U \ ~ 2.. <- if 5{\).t. f(c ") ~ f(~)