Page 174 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 12.3 Summary

 In this unit you have been introduced to the properties of continuous functions on bounded
closed intervals and you have seen the failure of these properties if the intervals are not
bounded and closed. These properties have been studied. It has been proved that if a
function f is continuous on a bounded and closed interval, then it is bounded and it also
attains its bounds. In the same section we proved the Intermediate Value Theorem that is
if f is continuous on an interval containing two points a and b, then f takes every value
between f(a) and f(b). The notion of uniform continuity is discussed. We have proved that
if a function f is uniformly continuous in a set A, then it is continuous in A. But converse is
not true. It has been proved that if a function is continuous on a bounded and closed
interval, then it is uniformly continuous in that interval. These properties fail if the intervals
are not bounded and closed. This has been illustrated with a few examples.

12.4 Keywords

Bounded Function: A function f continuous on a bounded and closed interval [a, b] is necessarily
a bounded function.
Boundedness: If f is a continuous function on the bounded closed interval [a, b] then there exists
points x and x in [a, b] such that f(x ) S f(x)  f(x ) for all x [a, b].
1 2 1 2
Intermediate Value Theorem: Let f be a continuous function on an interval containing a and b.
If K is any number between f(a) and f(b) then there is a number c, a  c S b such that f(c) = K.

12.5 Review Questions

1. Find the limits of the following functions:
1
(i) f(x) = x cos , x  0, as x  0.
x
x
(ii) f(x) = , x  0, as x  .
x
sin x
(iii) f(x) = , x  0, as x  .
x
2. For the following functions, find the limit, if it exists:
x - b
(i) f(x) = for x  b where b > 0, as x  b
x b
-
1
(ii) f(x) = - 1/x for x  0, as x  0
1 e
+

-
ì 3 x when x 1
(iii) f(x) = í as x  1.
>
î 2x when x 1
3. Test whether or not the limit exists for the following:
>
-
ì 3 x when x 1
ï
(i) f(x) = 1 when x = 1, as x  1.
í
ï
<
î 2x when x 1

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