Page 132 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 3. Test whether the following statements are true or false:
(i) The set Z of integers is not a NBD of any of its points.
(ii) The interval ]0, 1] is a NBD of each of its points
(iii) The set ]1, 3[  ] 4, 5[ is open.

(iv) The set [a,[  ] –, a] is not open.
(v) N is a closed set.
(vi) The derived set of Z is non-empty.
(vii) Every real number is a limit point of the set Q of rational numbers.

(viii) A finite bounded set has a limit point.
(ix) [4, 5]  [7, 8] is a closed set.
(x) Every infinite set is closed.

9.5 Summary

 In this unit, we have discussed various types of real functions. We shall frequently use
these functions in the concepts and examples to be discussed in the subsequent units
throughout the course.
 We have introduced the notion of an algebraic function and its various types. A function f:
S  R (S  R) defined as y = f(x), " x  S is said to be algebraic if it satisfies identically an
equation of the form
n
n–1
n–2
p (x) y + p (x) y + p (x) y + .... + p(x) y + p (x) = 0,
0 1 2 n
 where p (x), p (x), p (x) are polynomials in x for all x  S and n is a positive integer. In fact,
0 1 n
any function constructed by a finite number of algebraic operations—addition, subtraction,
multiplication, division and root extraction – is an algebraic function. Some of the examples
of algebraic functions are the polynomial functions, rational functions and irrational
functions.
 But not all functions are algebraic. The functions which are not algebraic, are called
transcendental functions. Some important examples of the transcendental functions are
trigonometric functions, logarithmic functions and exponential functions which have been
defined in this section. We have defined the monotonic functions also in this section.
 We have discussed some special functions. These are the identity function, the periodic
functions, the modulus function, the signum function, the greatest integer function, even
and odd functions. Lastly, we have introduced the bounded functions and discussed a few
examples.

9.6 Keywords

Upper Bound: Let f : SR be a function. It is said to be bounded above if there exists a real
number K such that f(x)  k " x S. The number K is called an upper bound of it.

Lower Bound: The function f is said to be bounded below if there exists a number k such that
f(x)  k x S. The number k is called a lower bound of f.
Cotangent Function: A function f : S  R defined.

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