Continuity at a Point and on an Open Interval
We noticed that the limit of a function as x approaches a
can often be found simply by calculating the value of the
function at a. Functions with this property are called
continuous at a.
In mathematics, the term continuous has much the same
meaning as it has in everyday usage. (A continuous
process is one that takes place gradually, without
interruption or abrupt change.)
Informally, to say that a function f is continuous at x = c
means that there is no interruption in the graph of f at c.
That is, its graph is unbroken at c and there are no holes,
jumps, or gaps.
2.3 Continuity
• 2.3.1 Continuity at a point
The definition says that f is
continuous at a if
f(x) approaches f(a) as
x approaches a. 2
A continuous function f
has the property that a small change in x produces only a
small change in f(x). In fact, the change in f(x) can be kept as
small as we please by keeping the change in x sufficiently
small.
3
If f is defined near a (in other words, f is defined on
an open interval containing a, except perhaps at a),
we say that f is discontinuous at a (or f has a
discontinuity at a) if f is not continuous at a.
Example 1 Figure 2
shows the graph of a
function f. At which
numbers is f
discontinuous?
Why?
SOLUTION It looks as if there
is a discontinuity when a = 1
because the graph has a break
there. The official reason that f
is discontinuous at 1 is that f(1)
is not defined.
Figure 1.25 identifies three values of x at which the graph of
f is not continuous. At all other points in the interval (a, b),
the graph of f is uninterrupted and continuous.
Figure 1.25
In Figure 1.25, it appears that continuity at x = c can be
destroyed by any one of the following conditions.
1. The function is not defined at x = c.
2. The limit of f(x) does not exist at x = c.
3. The limit of f(x) exists at x = c, but it is not equal to f(c).
If none of the three conditions above is true, the function f
is called continuous at c.
All polynomials are continuous for all real
numbers.
All rational functions are continuous for
every number in its domain. (i.e. f(x)=x/(x-
1) is not defined for x = 1, but it is continuous
for all other numbers)
The kinds of discontinuity:
1.
and exist ,
but x0 is not defined: Removable
x0 : Jump
2.
At least one of them does not exist,
If one of them is , x0 : Infinite
Examples:
y
y tan x
x o
2
x
2
y 1
y sin
x
0 x
x0
y Oscillating discontinuity point
x 1 o 1 x
x , x 1
(4) y f ( x) y
2 , x 1
1
1
1
lim f ( x) 1 f (1) 2
x 1
o 1 x
x 1
x 1 , x 0
y
(5) y f ( x) 0 , x 0 1
x 1 , x 0
o x
1
f (0 ) 1, f (0 ) 1
x0
The Intermediate Value Theorem
If f(x) is defined on closed interval [a,b] and
continuous on the open interval, (a,b), and if
f(a)<W< f(b), then there is at least one point,
c, between a & b, such that f(c) = W.
Corollary
There exists at least one point
Such that
y
y f (x)
a
o b x
Example – An Application of the Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the
polynomial function has a zero in the
interval [0, 1].
Solution:
Note that f is continuous on the closed interval [0, 1].
Because
it follows that f(0) < 0 and f(1) > 0.
You can therefore apply the Intermediate Value Theorem to
conclude that there must be some c in [0, 1] such that
Exercise:
There is at least one root in Interval [0,1]
Infinite Limits
Figure 1.40
Example 1 – Finding a Limit at Infinity
Find the limit:
Solution:
20
