Page 107 - DMTH401_REAL ANALYSIS

Unit 9: Functions

credited along with Isacc Newton for the invention of Calculus, Leibniz used the term function Notes
to denote a quantity connected with a curve. A Swiss mathematician, L. Euler [1707-1783] treated
function as an expression made up of a variable and some constants. Euler's idea of a function
was later generalized by an eminent French mathematician J. Fourier [1768-1830]. Another
German mathematician, L. Dirichlet (1805-1859) defined function as a relationship between a
variable (called an independent variable) and another variable (called the dependent variable).
This is the definition which, you know, is now used in Calculus.
The concept of a function has undergone many refinements. With the advent of Set Theory in
1895, this concept was modified as a correspondence between any two non-empty sets. Given
any two non-empty sets S and T, a function f from S into T, denoted as f: S  T, defines a rule
which assigns to each x  S, a unique element Leonard Euler y  T. This is expressed by writing
as y = f (x). This definition, as you will recall, was given in Section 1.2. A function f S  T is
said to be a

1. Complex-valued function of a complex variable if both S and T are sets of complex numbers;
2. Complex-valued function of a real variable if S is a set of real numbers and T is a set of
complex numbers;

3. Real-valued function of a complex variable if S is a set of complex numbers and T is a set
of real numbers;
4. Real-valued function of a real variable if both S and T are some sets of real numbers.

Since we are dealing with the course on Real Analysis, we shall confine our discussion to those
functions whose domains as well as co-domains are some subsets of the set of real numbers. We
shall call such functions as Real Functions.
In this unit, we shall deal with the algebraic and transcendental functions. Among the
transcendental functions, we shall define the trigonometric functions, the exponential and
logarithmic functions. Also, we shall talk about some special real functions including the bounded
and monotonic functions. We shall frequently use these functions to illustrate various concepts
in Blocks 3 and 4.

9.1 Algebraic Functions

In Unit 1, we identified the set of natural numbers and built up various sets of numbers with the
help of the algebraic operations of addition, subtraction, multiplication, division etc. In the
same way, let us construct new functions from the real functions which we have chosen for our
discussion. Before we do so, let us review the algebraic combinations of the functions under the
operations of addition, subtraction, multiplication and division on the real-functions.

9.1.1 Algebraic Combinations of Functions

Let f and g be any two real functions with the same domain S C R and their co-domain as the set
R of real numbers. Then we have the following definitions:
Definition 1: Sum and Difference of Two Functions
1. The Sum of f and g, denoted as f + g, is a function defined from S into R such that

(f + g) (x) = f(x) + g(x), If x  S.
2. The Difference of f and g, denoted as f – g, is a function defined from S to R such that
(f – g) (x) = f(x) – g(x), " x  S.

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