Page 161 - DMTH401_REAL ANALYSIS

Unit 11: Continuity

Notes
Example: Examine whether or not the function f: R ® R defined as,

ì 1, if x is irrational
f(x) = í
î 0, if x is rational
is totally discontinuous.
Solution: Let b be an arbitrary but fixed real number. Choose  = 1/2. Let  > 0 be fixed. Then the
interval defined by
|x – b| < 
is (x: b –  < x < b + )
or ]b – , b + [
This interval contains both rational as well as irrational numbers. Why?
If b is rational, then choose x in the interval to be irrational, If b is irrational then choose x in the
interval to be rational. In either case,
0 < |x – b|< 
and

|f(x) – f(b)| = I > .
Thus, f is not continuous at b. Since b is an arbitrary element of S, f is not continuous at any point
of S and hence is totally discontinuous.

There are certain discontinuities which can be removed. These are known as removable
discontinuities. A discontinuity of a given function f: S ® R is said to be removable if the limit
of f(x) as x tends to a exits and that
lim f(x)  f(a)
x® a
In other words, f has removable discontinuity at x = a if f(a+) = f(a–) but none is equal to f(a).
The removable discontinuities of a function can be removed simply by changing the value of the
function at the point a of discontinuity. For this a function with removable discontinuities can be
thought of as being almost continuous. We discuss the following example to illustrate a few
cases of removable discontinuities.

Example: Discuss the nature of the discontinuities of the following functions:
2
x - 4
(i) f(x) = , x  2
x - 4
= 1 x = 2
at x = 2.
(ii) f(x) = 3, x  3
= 1 x = 3
at x = 3.
2
(iii) f(x) = x , x  ] – 2, 0 (U) 0, 2 [
= 1 x = 0

at x = 0.

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