Page 155 - DMTH401_REAL ANALYSIS

Unit 11: Continuity

equal. If a function has these properties, then it is called a continuous function at the point a. We Notes
give the precise definition as follows:
Definition 1: Continuity of a Function at a Point

A function f defined on a subset S of the set R is said to be continuous at a point a  S, if
(i) lim f(x) exists and is finite
x a
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(ii) lim f(x) = f(a).
-
x a
Note that in this definition, we assume that S contains some open interval containing the point a.
If we assume that there exists a half open (semi-open) interval [a, c[ contained in S for some c  R,
then in the above definition, we can replace lim f(x) by lim f(x) and say that the function is
-
x a x a +
-
continuous from the right of a or f is right continuous at a.
Similarly, you can define left continuity at a, replacing the role of lim f(x) by lim f(x). Thus, f is
x a x a -
-
-
continuous from the right at a if and only if
f(a+) = f(a)
It is continuous from the left at a if and only if
f(a–) = f(a).
From the definition of continuity of a function f at a point a and properties of limits it follows
that f(a+) = f(a–) = f(a) if and only if, f is continuous at a. If a function is both continuous from the
right and continuous from the left at a point a, then it is continuous at a and conversely.
The definition X is popularly known as the Limit-Definition of Continuity.
Since lim f(x) is also defined, in terms of  and , we also have an equivalent formulation of the
-
x a
definition X. Note that whenever we talk of continuity of a function f at a in S, we always assume
that S contains a neighbourhood containing a. Also remember that if there is one such
neighbourhood there are infinitely many such neighbourhoods. An equivalent definition of
continuity in terms of  and  is given as follows:
Definition 2: (, )-Definition of 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.
Note that unlike in the definition of limit, we should have

|f(x) – f(a)|< E for |x – a| < 6.
The two definitions are equivalent. Though this fact is almost obvious, it will be appropriate to
prove it.

Theorem 1: The limit definition of continuity and the (, )-definition of continuity are equivalent.
Proof: Suppose f is continuous at a point a in the sense of the limit definition. Then given  > 0, we
have a  > 0 such that 0 < |x – a| <  implies |f(x) – f(a)| < . When x = a, we trivially have

|f(x) – f(a)| = 0 < .
Hence, |x – a| <   |f(x) – f(a)| < E
which is the (, )-definition.
Conversely we now assume that f is continuous in the sense of (, )-definition. Then for every
E > 0 there exists a  > 0 such that
|x – a| <   |f(x) – f(a)| < .

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