Continuity Chapter3
d Continuous -> FC EDVEso ISO FxED : 1x -
cl18 =
(f(x) f(c)kE -
t equivalent statements
2 : .
1
f is continuous at the point c ,
.
2
foreverysequence In in Dwith An =C ,
we have flan) f(c =
.
p (xf) /f g) 1.9) (fog) and I'
.
+ . .
are continuous
- Existence f(a) <0 <f(b) fis then there
of zeroes ->
,
cont .
is a cela b) , S .
th .
f(x) =
0
.
t Intermediate value thm f is cont .
1
.
If faf(b) ,
then
Stakes all values in
[f(ac J1b)] , ,
*
.
2
If f(a)> f(b) then stakes all values in [J(b) , , f(a)] .
t k For every (0 x) and KEX k
roots Yk
-
,
, , 1 ,
there existe a
unique Ye(0 , 8) S th . .
= .
x
y=
*
d k oot -
Unique Real number y .
d bounded -
If(x)KM for alle D
t boundedness - Every continuous function f [a bI R is bounded :
, .
t fila , b] Ris continuous Then the set.
fax[a bJB Las a ,
mascimum and a minimum .
FESO JSSO XyEDXxED
d
uniform continuity y/S 187 fly)k
- :
a -
= -
t
uniform continuity Every continuous function f : Ca bI,
R is
uniform continuous .
Differentiability Chapter 4
d limit point-CER is a limit point if ASSO JED 152 : /x-c/S
& limit -> VELo 750 ExeD( *cy (x : -
<(S =>
(f(x) L/x -
P JfL ,
↳Es th . .
f(x) = L ,
and hm f(x) =
Le then L L ,
=
P c ED is a limit point at D then , is continuous atc when if(x f(c) =
t 2 equivalent statements : 1 .
Ef(x) =
.
z
Forevery sequences in DIS With ni An =
c we have if(an) = L
PIff(x) h(x) (x) and if shem f(x) img(x) , ,
then if(x)limhas = hm g
f(x) f(x) -
d differentiability S is differentiable if the derivative a
exists .
t
If f is differentiable atc ,
then it is continuous ata
d continuously differentiable -
fis cont differentiable if it is differentiable and fis
.
cont
t ((g)'(x) g()/(g(x))2 = -
t f is increasing and differentiable atc then f'C 20 , .
L
fattains maximum minimum at and is differentiable then f'C
a or c ,
= o
t Rolle's theorem
of is cont.and differentiable Iff(a) =f(b) then there is ce(a b) s th f'( .
,
a , . .
= 0
.
t Mean value thm of is cont. and differentiable (a b) There exist a ceCa flSa on , .
,
b) s th
. .
t f'(x) =
0 , fis constant , f'/xo f is increasing , f' So fis stricly increasing
,
,
.
c Meanvalue inequality fis cont. diff There is . a constant C s th. .
If L then If flys C 2 y1
t (a b) and
fis cont and
diff on , af' . Then
L f is diff on a ,
with derivative
f'Cal =
L .
d Continuous -> FC EDVEso ISO FxED : 1x -
cl18 =
(f(x) f(c)kE -
t equivalent statements
2 : .
1
f is continuous at the point c ,
.
2
foreverysequence In in Dwith An =C ,
we have flan) f(c =
.
p (xf) /f g) 1.9) (fog) and I'
.
+ . .
are continuous
- Existence f(a) <0 <f(b) fis then there
of zeroes ->
,
cont .
is a cela b) , S .
th .
f(x) =
0
.
t Intermediate value thm f is cont .
1
.
If faf(b) ,
then
Stakes all values in
[f(ac J1b)] , ,
*
.
2
If f(a)> f(b) then stakes all values in [J(b) , , f(a)] .
t k For every (0 x) and KEX k
roots Yk
-
,
, , 1 ,
there existe a
unique Ye(0 , 8) S th . .
= .
x
y=
*
d k oot -
Unique Real number y .
d bounded -
If(x)KM for alle D
t boundedness - Every continuous function f [a bI R is bounded :
, .
t fila , b] Ris continuous Then the set.
fax[a bJB Las a ,
mascimum and a minimum .
FESO JSSO XyEDXxED
d
uniform continuity y/S 187 fly)k
- :
a -
= -
t
uniform continuity Every continuous function f : Ca bI,
R is
uniform continuous .
Differentiability Chapter 4
d limit point-CER is a limit point if ASSO JED 152 : /x-c/S
& limit -> VELo 750 ExeD( *cy (x : -
<(S =>
(f(x) L/x -
P JfL ,
↳Es th . .
f(x) = L ,
and hm f(x) =
Le then L L ,
=
P c ED is a limit point at D then , is continuous atc when if(x f(c) =
t 2 equivalent statements : 1 .
Ef(x) =
.
z
Forevery sequences in DIS With ni An =
c we have if(an) = L
PIff(x) h(x) (x) and if shem f(x) img(x) , ,
then if(x)limhas = hm g
f(x) f(x) -
d differentiability S is differentiable if the derivative a
exists .
t
If f is differentiable atc ,
then it is continuous ata
d continuously differentiable -
fis cont differentiable if it is differentiable and fis
.
cont
t ((g)'(x) g()/(g(x))2 = -
t f is increasing and differentiable atc then f'C 20 , .
L
fattains maximum minimum at and is differentiable then f'C
a or c ,
= o
t Rolle's theorem
of is cont.and differentiable Iff(a) =f(b) then there is ce(a b) s th f'( .
,
a , . .
= 0
.
t Mean value thm of is cont. and differentiable (a b) There exist a ceCa flSa on , .
,
b) s th
. .
t f'(x) =
0 , fis constant , f'/xo f is increasing , f' So fis stricly increasing
,
,
.
c Meanvalue inequality fis cont. diff There is . a constant C s th. .
If L then If flys C 2 y1
t (a b) and
fis cont and
diff on , af' . Then
L f is diff on a ,
with derivative
f'Cal =
L .
