ucsc
supplementary notes
ams/econ 11a
Continuous Functions, Smooth Functions and the
Derivative
c
2010 Yonatan Katznelson
1.
Continuous functions
One of the things that economists like to do with mathematical models is to extrapolate
the general behavior of an economic system, e.g., forecast future trends, from theoretical
assumptions about the variables and functional relations involved.
In other words, they
would like to be able to tell what’s going to happen next, based on what just happened;
they would like to be able to make long term predictions; and they would like to be able
to locate the extreme values of the functions they are studying, to name a few popular
applications.
Because of this, it is convenient to work with
continuous
functions.
Definition
1.
(i)
The function
y
=
f
(
x
)
is
continuous at
the point
x
0
if
f
(
x
0
)
is defined and
(1.1)
lim
x
→
x
0
f
(
x
) =
f
(
x
0
)
.
(ii)
The function
f
(
x
)
is
continuous in
the interval
(
a, b
) =
{
x

a < x < b
}
, if
f
(
x
)
is
continuous
at
every point in that interval.
Less formally, a function is continuous in the interval (
a, b
) if the graph of that function
is a
continuous
(i.e., unbroken) line over that interval.
Continuous functions are preferable to discontinuous functions for modeling economic
behavior because continuous functions are more predictable. Imagine a function modeling
the price of a stock over time.
If that function were discontinuous at the point
t
0
, then
it would be impossible to use our knowledge of the function up to
t
0
to predict what will
happen to the price of the stock after time
t
0
. On the other hand, if the stock price is a
continuous function of time, then we know that in the immediate vicinity of
t
0
, the prices
after
t
0
should be close to the prices up to and including
t
0
.
As nice as the behavior of continuous functions is, compared to functions that are not
continuous, it turns out that continuity by itself is not enough. Continuous functions can
also display ‘erratic’ behavior, and sudden changes in direction. What we would really like
to have are functions that change
smoothly
.
2.
Smooth curves
Consider the function
f
(
x
) =
11
x
(
x
2

x

2)
(
x

2)
4
+ 33
.
It is a continuous function, defined for all real numbers
x
, and its graph is shown below in
Figure 1.
1
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2
x0
y0
Figure 1.
The graph of
y
=
f
(
x
).
This graph certainly seems to be nice and smooth. But what does ‘smooth’ mean math
ematically? Smooth usually means no rough edges or corners. To see if we can translate
this into more mathematical terms, We’ll focus on the point (
x
0
, f
(
x
0
)) = (
x
0
, y
0
) on the
graph, and zoom in.
x0
y0
x0
y0
Figure 2.
Zooming in twice on (
x
0
, y
0
).
As you zoom in on something that is smooth, it begins to look flat (or straight) no matter
how curved it was to begin with. The figure on the lefthand side of Figure 2 looks much
straighter in the vicinity of the point (
x
0
, y
0
) than in the original graph. When we zoom in
again, this time on the circled portion of second graph, we get an arc that is almost straight
in the immediate vicinity of (
x
0
, y
0
), as illustrated in the graph on the righthand side of
Figure 2.
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 Spring '08
 QIAN
 Derivative, lim, Continuous function

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