Page 163 - DMTH401_REAL ANALYSIS

Unit 11: Continuity

3. Let f be a ............................... whose domain is a subset of the set R. The function f is said to Notes
be continuous at a point a if, for every sequence (x ) in the domain of f converging to a, we
n
have, lim f(X ) = f(a).
n ® ¥ n
4. Let f and g be two real functions such that the range of g is contained in, the domain of f.
If g is continuous at x = a, f is continuous at b = g(a) and ......................, for x in the domain
of g, then h is continuous at a.
5. A function f : S ® R fails to be ............................ on its domain S if it is not continuous at a
particular point of S.

11.4 Summary

The concept of the continuity of a function at a point of its domain and on a subset of its domain.
The limit definition and (, –)-definition of continuity. It has been proved that both the definitions
are equivalent. Sequential definition of continuity has been discussed and illustrations regarding
its use for solving problems have been given. The algebra of continuous functions is considered
and it has been proved that the sum, difference, product and quotient of two continuous functions
at a point is also continuous at the point provided in the case of quotient, the function occurring
in the denominator is not zero at the point. In the same section, we have proved that a continuous
function of a continuous function is continuous. Finally in Section 9.4, discontinuous and totally
discontinuous functions are discussed. Also in this section, one kind of discontinuity that is
removable discontinuity has been studied.

11.5 Keywords

Continuity: A function f is continuous at x = a if f is defined in a neighbourhood of a and
corresponding to a given number E > 0, there exists some number  > 0 such that |x – a | < 
implies |f(x) – f(a)| < E.
Sequential Continuity of a Function: Let f be a real-valued function whose domain is a subset of
the set R. The function f is said to be continuous at a point a if, for every sequence (x ) in the
n
domain of f converging to a, we have, lim f(X ) = f(a)
n ® ¥ n

11.6 Review Questions

1. Examine the continuity of the following functions:
(i) The function f: R – (0) ® R defined as

x
f(x) = ,
x
at the point x = 0
(ii) The function f: R ® (1) – R defined as
2
x - 1
f(x) = ,
x - 1
(iii) The function f: R – (0) – R defined as

1
f(x) = .
x

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