Page 175 - DMTH401_REAL ANALYSIS

Unit 12: Properties of Continuous Functions

2
x - 4 Notes
(ii) f(x) = , x R, as x  1.
2
x + 4
+
4 x - 2
(iii) f(x) = , x  O as x  0.
x
1 æ 1 2 ö
(iv) f(x) = ç - ÷ as x 1.
è
+
+
x 1 x 3 3x 5 ø
-
4. Discuss the continuity of the following functions at the points noted against each.
2
ì ï x for x  1
(i) f(x) = í as x  1.
ï î 0 for x = 1

<
ì 1 for 0 x 1
(ii) f(x) = í as x  1.
î 0 otherwise
2
x - 4
(iii) f(x) = when x  1.
x 1
-
f(1) = 2
as x  1.
+
ì ï (1 x) 1/x if x  0
(iv) f(x) = í as x  1.
ï î 1 if x = 0
ì 2 1
ï x sin if x  0
(v) f(x) = í x as x  1.
ï 1 if x = 0
î
5. Show that the function f : R  R defined as
1
1 |x|
+
does not attain its infimum.
6. Show that the function f : R  R such that

f(x) = x is not bounded but is continuous in [1, [.
7. Which of the following functions are uniformly continuous in the interval noted against
each? Give reasons.

(i) f(x) = tan x, x [0,/4J
1
(ii) f(x) = on [1, 4].
2
x - 3
Answers: Self Assessment

=
ì f(x) 1 if xis rational
1. (i) í
î = - 1 if x is irrational
=
ì f(x) x if xis rational
(ii) í
î = - x if x is irrational
the only point of continuity is 0.

LOVELY PROFESSIONAL UNIVERSITY 169