ANSWER :
a)
Find and open cover fo [0,1] that contains no finite subcover.
Take
Then take the cover
as
Then for any rational
so , is an open conver of .
Let is has an open subcover . i.e. there exist Such that
Take
Then by density property of Q there exist such that
Also,
So,
So, contains no finite subcover .
b)
Prove that the function
is continuous on [0,2] but not bounded on [0,2]
given,
Now f is discontinuous if
But
So,f is continuous on [0,2].
Take Then,by density property of Q.there exists rational number x such that
so,
Then
So,f(x) is not bounded
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