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MT2503 Multivariate Calculus Chapter 02: Functions of One Variable
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CHAPTER 02FUNCTIONS OF ONE VARIABLE
Definition
A function is rule that
assigns to each element sett and
a in ,
one
only
one element in
Seto
note that :
1) A and B need not be sets of real numbers
)
11 set A is called DOMAIN In one variable if ✗ c- A then flx) c- 0 The set of
,
.
.
all
possible values of fcx) is called RANGE of f
Domain
_ 0
Range
A 0
CONTINUITY
f- ( x ) is continuous at x=xo
L -
= Iim flx) ← limit from the
left if L +
=L -
=
L = f( Xo )
-
✗ → Xo
L
= Iim f- ( x ) c- limit from the
right
+
✗→ xot
If ( =/ L then
+
-
flx) is discontinuous
In most cases when [ + =L -
,
then flxo) =L
.
However this is not always true ,
consider :
fcx) = c-
not defined at x=0 but limit as ✗ → 0 does exist
In this case L + =L = 1
-
this is an example of in determinant form .
Limit can be found using L' Hospitals
rule the
or
by using power series for sink .
Function can be made continuous :
)={si¥
=/
fix ✗ o
= 0
1 x
?⃝
, DIFFERENTIABILITY
df
f- (x)
'
The derivative of C- ( x) ,
denoted Fx
or
,
at a
point ✗ ☐
is defined by the limit :
{ "?!}
df f'
1×0) f' (xo) lion
provided
= =
☒ "→ a.
←
that the limit exists
as x → scot and ✗ → xj
A function for which such a limit exists is said to be differentiable at x=xo
The derivative can also be defined as a function of x :
n→o{ }
df f "
Fx
=
f' ( x ) = 1in
If f- ( x) f- ( x) continuously
'
is continuous in a
given interval
,
then is a differentiable
function in that interval
•
f differentiable implies f continuous ( smoothness implies continuous)
differentiable ( continuous does not smoothness)
f continuous does not
imply f imply
•
2.5
Example
Is continuous and differentiable at x=o ?
glx)
{ It x2 x > 0
=
,
2
1- ✗ x < 0
,
← continuous
Iim
{ } lion
{ x2 }
'
ltx =L 1- =
I
.
✗ → ot ✗ → o
Thus ( +
=L -
=
1=910) ( since g defined
at x=o )
,
and so is continuous at -0
g x
✗ =D So flxth) =
f- ( h )
i
{ "-" } { }
L = lion =
"m -
h -
-
o
-
n→o- h so
-
-
4- =
n→o+{"+h }
' im =
n→o+{ }
lion h = O
e
differentiable
since L -
=L +
=
O
, g is differentiable at x=0
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