Page 162 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Solution:
(i) This function is discontinuous at x = 2. This is a removable discontinuity, for if we redefine
f(x) = 4, then we can restore the continuity of f at x = 2.
(ii) This is again a case of removable discontinuity at 3. Therefore, if f is defined by f(x) = 3 "
x  R, then it is continuous at x = 3.
(iii) This function is discontinuous at x = 0. Why? This is a case of discontinuity which is
removable. To remove the discontinuity, set f(0) = 0. In other words, define f as
2
f(x) = x , x  ] – 2, 0 [] 0, 2 [
= 0, x = 0

This is continuous at x = 0. Verify it.

Example: Let a function f: R ® R be defined as,
1
(i) f(x) = , x  0
x
= 0, x = 0

1
(ii) f(x) = , if x > 0
x
= 1, if x < 0

1
(iii) f(x) = , if x < 0
x
= 1, if x > 0
Test the continuity of the function. Determine the type of discontinuity if it exists.
Solution:

(i) Here f(0+) and f(0–) both do not exist (as finite real numbers) and so function is discontinuous.
This is not a case of removable discontinuity.

(ii) In this case, f(0) does not exist whereas f(0+) exists and f(0–) = f(0) = 1. This is not a case of
removable discontinuity.

Task Prove that the function f defined by f(x) = x sin 1/x if x  0 and f(0) = 1 has a
removable discontinuity at x = 0.
Self Assessment

Fill in the blanks:
1. A function f is said to be .......................... on a set S if, it is continuous at every point of the
set S. It is clear that a constant function defined on S is continuous on S.
2. A function f: S ® R is continuous at point a, in S if and only 31 far every sequence (x ),
n
(x  S) converging do a, f(x ) ..................... to f(a).
n n

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