Answer to Question #60675 in Real Analysis for esita

Question

3. a) Explain the order completeness property of R , and use it to show that the set






∈
+
= n N
n
n
S
1
has a supremum as well as infimum in R . (3)

b) Let f be the function defined by








+ ∈ ∞
∈
−
− ∈ ∞
=
1( 2 ) , if ,2[ [
, if ,1[ [2
3 2
2 ,1 if ] [1,
( )
2
2
x x
x
x
x
x x
f x
Discuss the continuity of f on [ ]− ∞, ∞ . (4)
c) Check whether the following sets are open, closed or neither:
i) ]6 ,1] [5 ∪ ,3[
ii) 




 ∪
7
10
,
4
3
,
9
5
]1,0[
iii) {5 n : n∈ N}