4.1 CONTINUITY OF A FUNCTION
Definition 4.1 :
The left-hand limit of a function f x , which is defined to the left
of a , at a point x a , exists and is equal to LL, written
LL
lim f x
x
f x
Let f x x / x for x R 0 and
f x 0
for x 0 . Note: f x sign x .
YES, (-1).
What is the left hand limit at x 0 ?
and what is the right hand limit at x 0 ? YES, ( 1).
NO.
Does the function have a limit at x 0 ?
1
0
x
-1
a
0 , however small, there exists some
if for any
L
f x L
, x , satisfying a
x a.
0 , such that
LL
f x
LL
for
x
a
,a
LL
f(x)
Definition 4.4 :
A function f x , which is defined on an open interval including the
point x a is continuous at that point if
lim f x
a
a±
Definition 4.2 :
x
x
The right-hand limit of a function f x , which is defined to the
R
right of a , at a point x a , exists and is equal to L , written
lim f x
x
LR
a
0 , however small, there exists some
if for any
R
f x L
, x , satisfying a x a
.
f x
Definition 4.3 : If xlim
a
at a point x
lim f x
x
L then the limit of a function
a
a equals to L:
lim f x
x
0 , such that
a
L
a
f a
THEOREM 4.1 : Suppose that f(x) and g(x) are continuous functions and
that c LVFRQVWDQW7KHIROORZLQJDUHDOVRFRQWLQXRXV
(i)
(ii)
(iii)
(iv)
(v)
(vi)
f(x) = x for x
c f(x)
f(x) + c
f(x) ± g(x)
f(x) g(x)
f(x) / g(x) for g(x) 
x=f ±1(y) if it exists
R is continuous. Is f (x)/f (x), x
R also continuous?
No, (we cannot use the Theorem 4.1 as f (x)=0 for x=0.)
g(x) = 2x for x
R++ is continuous. Is 1/g (x) for x
R++ also
continuous?
Yes, we can use the Theorem 4.1 as g(x)  x R++.)
2
4
Definition 4.5 : If the value of a function f x , x R , becomes unbounded as x
approaches some value x a either from the left or right, then we
say the line x a is an asymptote of the function.
lim f x
x
then the line x
a
wf
Definition 4.6 : Let function f x , x R , be defined on a closed interval a,b :
(i) f x is continuous from the right at the point x a if its
a exists and equals to f a ;
(ii) f x is continuous from the left at the point x
b if its
left-hand limit at x b exists and equals to f b ;
(iii) f x is continuous on the closed interval a,b if it is
continuous at every point inside a,b and it is
Graph and explain where the function
f x
x 2 / x 2 x 2 is discontinuous.
Q continuous (4.1 vi.)
Q continuous (4.1 i.)
Q
C Q
c0 wf
1
Q continuous (4.1 ii.).
pQ continuous (4.1 i)
pQ C Q continuous (4.1 iii. and C Q also continuous)
Consider the Bertrand model of price competition. Two firms, A and B, set prices pA
and pB, respectively. The firm offering the lower price captures the entire market. If
the prices are the same then they share the market equally.
Assume that demand function is Q = 20 ± p,
each firm faces equal cost function C(q i) = 2 qi.
(a) If firm B sets a price pB ILQGDQGJUDSK$¶VUHYHQXHRA(pA, pB)|pB=20, costs
CA(pA, pB)|pB=20 and profit A(pA, pB)|pB=20 functions.
(b) Re-do profit function graph with pB = 6 and explain why the functions are
discontinuous. Explain why A prefers to undercut B.
if p A pB
0
R A p A , pB
0
1
(a) pB=20
f x
±2 ±1
f
(c) Re-do profit function graph with pB = 1 and explain why A prefers to set pA 2.
(d) Re-do profit function graph with pB = 2 and explain pA = pB = 2 is the
equilibrium price.
continuous from the right and left at points a & b.
It is discontinuous at two points :
x = ±1 (undefined &vertical asymptote) and
x = 2 (undefined)
1
a is a
vertical asymptote.
right-hand limit at x
f L continuous
Q L continuous
In such cases the function is not continuous.
f x
If xlim
a
Q L
x
2
R
100
if p A pB
0
20 p A p A / 2 if p A pB
20 p A p A
if p A pB
C A p A , pB
C
if p A pB
0
20 p A
if p A pB
20 p A 2 if p A pB
A p A , pB
20 p A p A 2 / 2 if p A pB
20 p A p A 2
if p A pB
81
ʌ
40
Given the production function Q L bL,b 0
q , for a
defined on R+, derive the cost function, C q , and the profit function,
perfectly competitive firm. Let c0 be fixed costs and w be the unit price of labor L.
Prove that all three functions are continuous. Use Definition 4.4 for Q L and
Theorem 4.1 for C q and q .
Q L
bL continuous at x:
:
s.t.
:
s.t.
:
Q L
Q x
for L s.t. L x
bL bx
for L s.t. L x
L x
s.t.
6
b
for L s.t. L x
b
.
0
10
20 pA
(b) pB=6
ʌ
56
28
0
-40
20 pA
0
0
-40
2
(c) pB=1
(d) pB=2
ʌ
ʌ
11
20 pA
1
2
6
pA
0
-19
-40
[1,-9½]
8
pA
0
-40
2
pA
4.2 INTERMEDIATE VALUE THEOREM
THEOREM 4.2 : (INTERMEDIATE VALUE THEOREM)
Suppose that f(x) is a continuous function on a closed
interval [a,b] and that IDIE. Then for any number y
between f(a) and f(b), there is some value of , say x = c,
between a and b such that y = f(c).
THEOREM 4.3 : If the demand and supply functions are continuous and
the following two conditions are satisfied:
(i) at zero price, demand exceeds supply, D(0)±S(0) = z(0) >0
(ii) there exists some price, p>0, at which supply exceeds
demand, meaning that z(p) < 0
then there exists a positive equilibrium price in the market.
&RQVLGHUWKHIROORZLQJH[DPSOHRI+RWHOOLQJ¶VORFDWLRQPRGHO(DFKRIWKH
two firms sell a homogeneous product and charges a price of $25 while facing
constant unit cost of $15. There are N = 1000 consumers who are uniformly
distributed along a street one mile in length, represented by the unit interval
[0, 1]. The two firms, A and B, will choose locations, LA and LB respectively,
on the line [0, 1] in such a way as to maximize market share, and hence profit.
(a) For any general location of the firm B, LB òILQGDQGJUDSK$¶V
PDUNHWVKDUHDQG$¶VSURILWIXQFWLRQ$UHWKH\GLVFRQWLQXRXV":KDW
should A do to increase its profit?
(b) For any general location of the firm B, LB !òILQGDQGJUDSK$¶V
profit function. What should A do to increase its profit?
(c) For location of the firm B in the middle, LB òILQGDQGJUDSK$¶V
profit function. What should A do to increase its profit?
(d) Where should firm A locate? Where should firm B locate?
Consider D(p) = 50 ± 2p and S(p) = -10 + p
Graph D(p) and S(p) on one diagram and z(p) on another. Use Theorem 4.3 to argue
that equilibrium price exists and find it.
P
P
S(p)
20
D(p) continuous
S(p) continuous
z(0) = 60 > 0
z(30) = ± 30 < 0
20
D(p
±10 0 10
z(p) = 60 ± 3p
50 Q
60 Q
0
Let D(p) = 100 ± 2p for p>45 and D(p) = 120 ± 2p otherwise. Let S(p) = -20 + p.
Graph D(p) and S(p) on one diagram and z(p) on another. Use Theorem 4.3 to argue
that equilibrium price does not exist (what condition is absent).
p
D(p)
S(p)
z(p) =
45
50
45
±20 0 10 25 30
p
120
Q
15 0 5
10
± 3p
{140
120 ± 3p
140
Q
if p 45
otherwise
D(p) NOT continuous x
(a) Note that the profit is given by the market share: Each customer increases the profit
by $10 (= $25 ± $15).
$¶V Market
LB
From the graphs below we can see that the
Share
equilbrium location for both firms is exactly in
½
LB
the middle of the one mile street.
2
1±LB
Both firms take ½ of the market.
1±LB
The 500 customers of each firm generate $5,000
2
profit for each firm.
0
LB
1
$¶V3URILW
0
LB > ½ 1
(b) Firm A should move
to the left of firm B
$¶V3URILW
0
LB = ½
(d) Firm A should move
to where firm B is.
12
$¶V3URILW
0
LB < ½
1
(c) Firm A should move
to the right of firm B