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Unformatted text preview: gain, transforming the original function will help us out: 2 x + 5 ( 2 x + 2) + 3
x +1
3
3
f ( x) =
=
=2
+
=2+
x +1
x +1
x +1 x +1
x +1 3 lim 2 + =2
x→
+∞
x +1 17 Continuous Functions
Definition A function y = f ( x ) is said to be continuous at point x0 if it
satisfies the following requirements:
satisfies
y
1) x0 is in the domain of the function
1) x0 = 0 1
y=
x is not in the domain of 1 x 2) There exists a limit
2) 0, x = 0
f ( x) = 1, x ≠ 0 0 lim f ( x ) = f ( x0 ) x → x0 lim f ( x ) exists, but it’s not
x→ 0 equal to f ( 0) x y
1
0 x 18 Continuity and Smoothness
Definition: A function is said to be continuous in set S if it is
continuous at each point of S.
continuous
Continuity in a particular set has little to do with smoothness in that set.
This function is continuous,
This
is
but it is not smooth (so we
know it won’t have limit
values defined at its kinky
points).
points). It turns out that if we can differentiate a function in a particular set, it will
It
also be smooth in that set.
in
19 Differentiable Functions
Definition: a function is differentiable at point x0 if it has a
lim
derivative at that point, i.e. if the limit ∆x →0 f ( x0 + ∆x ) − f ( x0 )
exists.
∆x Definition: a function is differentiable in set S if it is differentiable at
every point of S.
every
Definition: a function is continuously differentiable in set S if its
( 1)
derivative in that set is a continuous function (notation: f ∈ C , f ∈ C ′ ) ⇒
Continuity ⇐
/ Differrentiability lim ∆x →
0 f ( x0 + ∆x ) − f ( x0 ) ≡ { ∆x ≠ 0} = f ( x0 + ∆x ) − f ( x0 )
∆x
∆x ( ) lim f ( x0 + ∆x ) − f ( x0 ) = f ′( x0 ) lim ∆x = 0 ∆x →0 ⇓ ∆x →0 lim f ( x0 + ∆x ) = f ( x0 ) ∆x →0 20 Economic Functions
The many types of economic functions typically used include
1)Profit functions
2)Cost functions
3)Production functions
4)Demand functions
5)Supply functions The overwhelming majority of the functions actually used in the
The
economic analysis are assumed to be differentiable everywhere in a
specific domain (often all real numbers). That, of course, implies
that these functions are also assumed to be continuous everywhere
in the domain.
in 21 Which Statement is True?
1. Continuity at a point implies
1.
differentiability
differentiability
2. Differentiability at a point does not require
2.
this point to be in the function’s domain
this
3. Differentiability at a point requires
3.
continuity
continuity
4. Continuity at a point allows for different
4.
left and rightside limits
left22...
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 Fall '11
 Kim

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